STRUCTURAL DYNAMICS: RECENT ADVANCES
Proceedings of the 6th International Conference
Volume II
Proceedings of the Sixth International Conference on Recent Advances in
Structural Dynamics, held at the Institute of Sound and Vibration Research,
University of Southampton, England, from 14th to 17th July, 1997, co-sponsored by
the US Airforce European Office of Aerospace Research and Development and the
Wright Laboratories, Wright Patterson Air Force Base.
Edited by
N,S. FERGUSON
Institute of Sound and Vibration Research,
University of Southampton, Southampton, UK.
H.F. WOLFE
Wright Laboratory,
Wright Patterson Air Force Base, Ohio, USA.
and
C MEI
Department of Aerospace Engineering,
Old Dominion University, Norfolk, Virginia, USA.
© The Institute of Sound and Vibration Research, University of Southampton, UK.
ISBN no. 0-85432-6375
19970814 055
VV T.- , ..
PREFACE
The papers contained herein were presented at the Sixth International
Conference on Recent Advances in Structural Dynamics held at the Institute of
Sound and Vibration Research, University of Southampton, England in July 1997.
The conference was organised and sponsored by the Institute of Sound and
Vibration Research and co-sponsored by the Wright Laboratories, Wright Patterson
Air Force Base. We wish to also thank the following for their contribution to the
success of the conference: the United States Air Force European Office of Aerospace
Research and Development. The conference follows equally successful conferences
on the same topic held at Southampton in 1980, 1984,1988,1991 and 1994.
There are over one hundred papers written by authors from approximately
20 different countries, making it a truly international forum. Many authors have
attended more than one conference in the series whilst others attended for the first
time.
It is interesting to note the change in emphasis of the topics covered.
Analytical and numerical methods have featured strongly in all the conferences.
This time, system identification and power flow techniques are covered by even
more papers than previously. Also, there are many contributions in the field of
passive and active vibration control. Papers dealing with nonlinear aspects of
vibration continue to increase. These observations seem to reflect the trend in
current research in structural dynamics. We therefore hope that the present series
of International Conferences will play a part in disseminating knowledge in this
area.
We would like to thank the authors, paper reviewers and session chairmen
for the part they played in making it a successful conference.
My personal thanks go to the following individuals who willingly and
enthusiastically contributed to the organisation of the event:
Dr. H.F. Wolfe Wright Laboratories, WPAFB, USA
Dr. C. Mei Old Dominion University, USA
Mrs. M.Z. Strickland ISVR, University of Southampton, UK
Grateful thanks are also due to many other members of ISVR who contributed to
the success of the event.
N.S. Ferguson
Sixth International Conference on
Recent Advances in Structural Dynamics
Volume II
Contents
Page No.
INVITED PAPER
R.D. BLEVINS
On random vibration, probability and fatigue 881
ACOUSTIC FATIGUE I
58. J. LEE and K.R. WENTZ
Strain power spectra of a thermally buckled plate in
random vibration 903
59. S.A. RIZZI and T.L. TURNER
Enhanced capabilities of the NASA Langley thermal
acoustic fatigue apparatus 919
60. I. HOLEHOUSE
Sonic fatigue characteristics of high temperature materials
and structures for hypersonic flight vehicle applications 935
61 . M. FERMAN and H.F. WOLFE
Scaling concepts in random acoustic fatigue 953
ACOUSTIC FATIGUE II
62. H.F. WOLFE and R.G. WHITE
The development and evaluation of a new multimodal
acoustic fatigue damage model 969
63. B. BENCHEKCHOU and R.G. WHITE
Acoustic fatigue and damping technology in composite
materials 981
64. D. MILLAR
The behaviour of light weight honeycomb sandwich panels
under acoustic loading 995
65. P.D. GREEN and A. KILLEY
Time domain dynamic Finite Element modelling in acoustic
fatigue design 1007
SYSTEM IDENTIFICATION II
66.
U. PRELLS, A.W. LEES, M.I. FRISWELL and M.G. SMART
Robust subsystem estimation using ARMA-modelling in
the frequency domain
1027
67.
Y.Q. NI, J.M. KO and C.W. WONG
Mathematical hysteresis models and their application to
nonlinear isolation systems
1043
68.
M.G. SMART, M.I. FRISWELL, A.W. LEES and U. PRELLS
The identification of turbogenerator foundation models
from run-down data
1059
69.
C. OZTURK and A. BAHADIR
Shell mode noise in reciprocating refrigeration
compressors
1073
70.
T.H.T. CHAN, S.S. LAW and T.H. YUNG
A comparative study of moving force identification
1083
71.
P.A. ATKINS and J.R. WRIGHT
Estimating the behaviour of a nonlinear experimental multi
degree of freedom system using a force appropriation
approach
1099
POWER FLOW TECHNIQUES II
72.
R.S. LANGLEY, N.S. BARDELL and P.M. LOASBY
The optimal design of near-periodic structures to minimise
noise and vibration transmission
1113
73.
J.L. HORNER
Effects of geometric asymmetry on vibrational power
transmission in frameworks
1129
74.
M. IWANIEC and R. PANUSZKA
The influence of the dissipation layer on energy flow in
plate connections
1143
75.
H. DU and F.F. YAP
Variation analysis on coupling loss factor due to the third
coupled subsystem in Statistical Energy Analysis
1151
76.
S.J. WALSH and R.G. WHITE
The effect of curvature upon vibrational power
transmission in beams
1163
77.
S. CHOI, M.P. CASTANIER and C. PIERRE
A parameter-based statistical energy method for mid¬
frequency vibration transmission analysis
1179
PASSIVE AND ACTIVE CONTROL III
78. Y. LEI and L. CHEN
Research on control law of active suspension of seven
degree of freedom vehicle model 1195
79. M. AHMADIAN
Designing heavy truck suspensions for reduced road
damage 1203
80. A.M. SADRI, J.R. WRIGHT and A.S. CHERRY
Active vibration control of isotropic plates using
piezoelectric actuators 1217
81. S.M. KIM and M.J. BRENNAN
Active control of sound transmission into a rectangular
enclosure using both structural and acoustic actuators 1233
82. T.J. SUTTON, M.E. JOHNSON and S.J. ELLIOTT
A distributed actuator for the active control of sound
transmission through a partition 1247
83. J. RO, A. A-ALI and A. BAZ
Control of sound radiation from a fluid-loaded plate using
active constraining layer damping 1257
ANALYTICAL METHODS 11
84. E. MANOACH, G. DE PAZ, K. KOSTADINOV and
F. MONTOYA
Dynamic response of single-link flexible manipulators 1275
85. B. KANG and C.A. TAN
Wave reflection and transmission in an axially strained,
rotating Timoshenko shaft 1291
86. Y. YAMAN
Analytical modelling of coupled vibrations of elastically
supported channels 1329
87. R.S. LANGLEY
The response of two-dimensional periodic structures to
harmonic and impulsive point loading 1345
NONLINEAR VIBRATION III
88. H. OYANG, J.E. MOTTERSHEAD, M.P. CARTMELL and
M.L FRISWELL
Stick-slip motion of an elastic slider system on a vibrating
disc
1359
89.
R.Y.Y. LEE, Y. SHI and C. MEI
A Finite Element time domain multi-mode method for
large amplitude free vibration of composite plates
1375
90.
P. RIBEIRO and M. PETYT
Nonlinear forced vibration of beams by the hierarchical
Finite Element method
1393
91.
K.M. HSIAO and W.Y. LIN
Geometrically nonlinear dynamic analysis of 3-D beam
1409
92.
R.Y.Y. LEE, Y. SHI and C. MEI
Nonlinear response of composite plates to harmonic
excitation using the Finite Element time domain modal
method
1423
93.
C.W.S. TO and B. WANG
Geometrically nonlinear response analysis of laminated
composite plates and shells
1437
ANALYTICAL METHODS III
94.
R.S. HWANG, C.H.J. FOX and S. McWILLIAM
The free, in-plane vibration of circular rings with small
thickness variations
1457
95.
D.J. GORMAN
Free vibration analysis of transverse-shear deformable
rectangular plates resting on uniform lateral elastic edge
support
1471
96.
R.G. PARKER and C.D. MOTE, Jr.
Wave equation eigensolutions on asymmetric domains
1485
97.
A.V. PESTEREV
Substructuring for symmetric systems
1501
RANDOM VIBRATION I
98.
G.FUandJ. PENG
Anaytical approach for elastically supported cantilever
beam subjected to modulated filtered white noise
1517
99.
S.D. FASSOIS and K. DENOYER
Linear multi-stage synthesis of random vibration signals
from partial covariance information
1533
100.
CW.S.TOand Z. CHEN
First passage time of multi-degrees of freedom nonlinear
systems under narrow-band non-stationary random
excitations
1549
101.
C. FLORIS and M.C. SANDRELLI
Random response of Duffing oscillator excited by quadratic
polynomial of filtered Gaussian noise
1565
102.
S. McWILUAM
Extreme response analysis of non-linear systems to random
vibration
1581
103.
M. GHANBARI and J.F. DUNNE
On the use of Finite Element solutions of the FPK equation
for non-linear stochastic oscillator response
1597
RANDOM VIBRATION II
104.
T.L. PAEZ, S. TUCKER and C. O’GORMAN
Simulation of nonlinear random vibrations using artificial
neural networks
1613
105.
D.Z. LI and Z.C. FENG
Dynamic properties of pseudoelastic shape memory alloys
1629
106.
Z.W. ZHONG and C. MEI
Investigation of the reduction in thermal deflection and
random response of composite plates at elevated
temperatures using shape memory alloys
1641
SIGNAL PROCESSING I
107.
M. FELDMAN and S. BRAUN
Description of non-linear conservative SDOF systems
1657
108.
N.E. KING and K. WORDEN
A rational polynomial technique for calculating Hilbert
transforms
1669
109.
D.M. LOPES, J.K. FIAMMOND and P.R. WHITE
Fractional Fourier transforms and their interpretation
1685
SYSTEM IDENTIFICATION III
no. J. DICKEY, G. MAIDANIK and J.M. D’ARCHANGELO
Wave localization effects in dynamic systems 1701
111. P. YUAN, Z.F- WU and X.R. MA
Estimated mass and stiffness matrices of shear building
from modal test data
1713
112. YU. I. BOBROVNirSKn
The problem of expanding the vibration field from the
measurement surface to the body of an elastic structure 1719
113 M. AMABILI and A. FREGOLENT
Evaluation of the equivalent gear error by vibrations of a
spur gear pair
1733
ON RANDOM VIBRATION, PROBABILITY, AND FATIGUE
R. D. Blevins
Rohr Inc., Mail Stop 107X
850 Lagoon Drive
Chula Vista, California 91910
ABSTRACT
Analysis is made to determine the properties of a random process consisting of the
sum of a series of sine waves with deterministic amplitudes and independent, random
phase angles. The probability density of the series and its peaks are found for an arbitrary
number of terms. These probability distributions are non-Gaussian. The fatigue resulting
from the random vibration is found as a function of the peak-to-rms ratio.
1. INTRODUCTION
Vibration spectra of aircraft components often are dominated by a relatively small
number of nearly sinusoidal peaks as shown in Figure 1. The time history of this process,
shown in Figure 2, is irregular but bounded. The probability density of the time history,
shown in figure 3 only roughly approximates a Gaussian distribution and it does not exceed
2.5 standeird deviations.
The time history of displacement or stress of these processes over a flight or a take
off time can be expressed as a Fourier series of a finite number of terms over the finite
sampling period T.
N
y = 0,nCOs(u)-ntn + <^n), 0 < in < T, Un > 0 (l)
n=l
Each frequency Un is a positive, non-zero integer multiple of 27r/r. The following model is
used for the nature of the Fourier series: 1) the amplitudes a-n are positive and deterministic
in the sense that they do not vary much from sample to sample, 2) the phases (j>n are random
in the sense that they vary from sample to sample, they are equally likely to occur over
the range -oo < 0n < oo. This last condition implies that the terms on the right hand
side of equation (1) are statistically independent of each other.
We can generate an ensemble of values of the dependent variable Y by randomly
choosing M sets of N phase angles = 1,2..^), computing Y at some ffxed time
from equation (1), choosing another set of phases, computing a second value of Y and so
881
on until we have a statistically significant sample of M Y's. This random phase approach,
introduced by Rayleigh (1880), models a multi-frequency processes where each frequency
component is independent and whose power spectral density (PSD) is known.
The maximum possible (peak) value of equation (1) is the sum of the amplitude of
each term (recall > 0). The mean square of the sum of independent sine waves is the
sum of the mean squares of the terms.
N
^peak ~ 'y ^
n=l
= Na, for ai= 02= an = CL
(2a)
(2&)
X N N
Yrms = ^ f [Y anCO«(2wt„/T + <Pri)?dtn = 5
= |iVa^, for ai = 02 = an = a
The peak-to-rms ratio of the sum of N mutually independent sine waves thus is,
N N
I rms
1/2
n=l 71=1
= (2A/')^/^, for Oi = 02 = an = o.
(3a)
{Sb)
(4a)
(46)
Equation (4b) shows that the peak-to-rms ratio for an equal amplitude series increases from
2^/^ for a single term (N=l) and approaches infinity as the number of terms N approaches
infinity, as shown in Figure 4. The probability of Y is zero beyond the peak value. For
example, there is no chance that the sum of any four {N = 4) independent sinusoidal terms
will be greater than 8^/^ = 2.828 times the overall rms value.
2. PROBABILITY DENSITY OF A SINE WAVE
The probability density py (y) of the random variable Y is probability that the random
variable Y has values within the small range between y and y -b dy, divided by dy. p(Y)
has the units of 1/Y. Consider single a sine wave of amplitude a^, circular frequency uJri)
and phcLse
Y = On COs{0Jntn + 0 < < 277. (5)
Y is the dependent random variable. The independent random variables are tn or <l>n- The
probability density of a sine wave for equal likely phases p((l>n) = l/(27r), or equally likely
times, p[tn) = 1/T, is (Bennett, 1944; Rice 1944, art. 3.10),
wiy) =
77 ^(a^ - j/2) 1/2^ if <y <an\
0,
if I2/I > CLn
(6)
882
The probability density of the sine wave is symmetric about y = 0, i.e., pyiv) ~ PYi~y))
it is singular at y = Cn, and it falls to zero for jyl greater than an as shown in Figure 5.
The characteristic function of a random variable x is the expected value of
C{f) = r (7)
J —CO
and it is also the Fourier transform of the probability density function (Cramer, 1970,
pp. 24-35; Sveshnikov, 1965; with notation of Bendat, 1958). j = is the imaginary
constant. The characteristic function of the sine wave is found using equations (13) and
(14) and integrating over the range 0 < X < a^. (Gradshteyn, Ryzhik, Jeffrey, 1994, article
3.753).
Cn(f) = 2(7ra„)-‘ T" cos{27rfY))[l - {Y / dY = Jo(27r/a„), (8)
Jo
The characteristic function of a sine wave is a Bessel function of the first kind and zero
order (Rice, 1944, art. 3.16). Equations (6) and (8) are starting points for determining
the probability density of the Fourier series.
3. PROBABILITY DENSITY OF THE SUM OF N SINE WAVES
It is possible to generate an expression for the probability density of Fourier series
(equation l) with 1,2, 3, to any number of terms provided the sine wave terms are mutually
independent. This is done with characteristic functions. The characteristic function of the
sum of N mutually independent random variables (Y = Xi -j- X2 + -■■i- Xj\/) is the product
of their characteristic functions (Weiss, 1990, p.22; Sveshnikov, pp. 124-129),
c(/) = r .. r e^2-/{^.+^=+-
7 —00 7—00
N ^00 N
= n / = n CM)- (9)
The symbol 11 denotes product of terms. The characteristic function for the sum of N
independent sine waves is found from equations (8) and (9).
C{f)^
n!Li *^0 (27r/an) , unequal an
[Jo(27r/a)]^, ai == a2 = = a
(10)
The probability density of Y is the inverse Fourier transform of its characteristic function
(Sveshnikov, 1968, p. 129).
py{y) = r e-^^-fyc{f)df
7—00
(11)
883
By substituting equation (10) into equation (11) we obtain an integral equation for the
probability density of a N-term finite Fourier series of independent sine waves (Barakat,
1974).
.oo N
Pviy) = 2 / cos{2Tryf) { TT Jo(27r/an) } df,
''0 n=l
iV = 1,2,3...
(12)
If all N terms of the Fourier series have equal amplitudes a = ai — a2 = On = a/sr, then
this simplifies,
py(y) = 2 r cos(2iryf)[Jo{2wfa)fdf, N = 1,2,3... (13)
Jo
These distributions are symmetric about y — 0 as are all zero mean, sum-of-sine-wave
distributions. Figures 5 and 6 show results of numerically integrating equations (45) and
(46) over interval / = 0 to / = 15a using Mathematica (Wolfram, 1995).
Barakat (1974, also see Weiss, 1994, p. 25) found a Fourier series solution to equation
(45). He expanded the probability density of the N term sum in a Foui'ier series over the
finite interval -Ly < Y < Ly where Ly = ai + a2 + - + The result for unequal
amplitudes is ,
\v\<Ly.
i=L n=l
For equal amplitudes, ai = a2 = a,^ = o, Ly = Na, and
i=l
\y\ < Na.
(14)
(15)
Figure 6 shows that the Fourier series solution (equation 15) carried to 20 terms to be
virtually identical to numerical integration of equation (13) and it compares well with the
approximate solution. Note that theory requires py{\yT\ > Ly) = 0.
A power series solution for equation (13) can be found with a technique used by Rice
(1944, art. 16) for shot noise and by Cramer (1970) who called it an Edgeworth series. The
Bessel function term in equation (13) is expressed as an exponent of a logarithm which is
then expanded in a power series,
[Jo(27r/a)]-^ = ex'p{N ln[Jo[2'Kaf)\), (16)
= exvi-Nir'^a^f - (l/4)iV7r^o'‘/'' - (l/9)N-K^a^f + (n/192)Arx5a®/-)-
= + -1
Substituting this expansion into equation (13) and rearranging gives a series of integrals,
which are then solved (Gradshteyn, Ryzhik, Jeffrey, 1994, arts. 3.896, 3.952) to give a
884
power series for the probability density of the equal-amplitude N-term Fourier series sum.
VY{y) =
g^iFi[-2,V2,yV(2yA.,)l
-(
11
r(9/2)
(17)
192iV3 32iv2^ ^1/2
)^^iFi[-4,l/2,yV(2i;L.)l + -)^ \y\<Na
'PY{\y\ > = 0 and Yrms is given by equation (3b). There are two special func¬
tions in equation (17), the gamma function T and the confluent hypergeometric function
iFi[n,'y,z]. These are defined by Gradshteyn, Ryzhik, and Jeffrey (1994).
As N approaches infinity, the peak-to-rms (equation 4b) ratio approaches infinity, and
equation (51) approaches the normal distribution,
\im pY{y) = ~;^ — . (18)
N-*oo V^Yrms
as predicted by the central limit theorem (Cramer, 1970; Lin, 1976).
4. PROBABILITY DENSITY OF PEAKS
Theories for calculating the fatigue damage from a time history process generally
require knowledge of the peaks and troughs in the time history. This task is made simpler
if we assume that the time history is narrow band. If Y{t) is narrow band that is, that
each trajectory of Y{t) which crosses zero has only a single peak before crossing the cixis
again, then (1) the number of peaks equals the number of times the time history crosses
the axis with positive slope, and (2) only positive peaks occur for Y{t) >0 and they are
located at points of zero slope, dY{t)ldt = 0. Lin (1967, p. 304) gives expressions for the
expected number of zero crossings with positive slope (peaks above the axis) per unit time
for a general, not necessarily narrow band, process,
ElNo+]= f ypyy(0,y)<iy (19)
Jo
and the probability density of the peaks for a narrow band process.
= (20)
In order to apply these expressions, the joint probability distribution of Y and Y must be
established. The joint probability density function Pyriy^y) random variable
Y and Y is the probability that Y falls in the range between y and y + dy and y falls in the
range between y and y-\-dy, divided by dydy. The derivative of the sine wave Y (equation
12) with respect to time can be expressed in terms of Y ,
dY/dt = Y — -Gn^n sin{u)nt + 0n) == ~ Y^, \Y\ < On. (21)
885
The joint probability density is the inverse Fourier transform of its characteristic
function,
/CO poo
/ (30)
-oo J—oo
The proof of equations (28), (29), and (30) can be found in Chandrasekhar (1943), Willie
(1987), Weiss and Shmueli (1987), and Weiss (1994, pp. 21-26).
Since the probability is symmetric about y — y = 0, Pyriv^v) ~ Pyy(~S/j“y)) only
symmetric terms survive the integration. Substituting, equation (29) into equation (30)
and expanding gives and integral expression for the joint probability of y and Y.
Pyy(2/,y)== [ [ {JJ ^o(27ranY^/f + /|cj2)}cos(27r/iy)cos(27r/2y)d/i<i/2 (31)
It is also possible to expand the joint probability of Y and Y in as double finite Fourier
series. The result is:
.. oo oo N I . 7
Pyriv^y) = (^)2j}cos(i7ry/Ly)cos(/:7ry/L^)
aik = 1, i, /c> 0; 1/2, i — Qork = 0; 1/4, k = 0
(32)
(33)
The expected number of peaks per unit time and the probability distribution of the narrow
band peaks is obtained by substituting this equation into equations (19) and (20) and
integrating. The results are:
n—1
^0 k=0
TTUr
Ly ‘
(34)
y OO OO iV
:/-)' + (^P)}sin(i^A/Ly)
JuY J^Y
where
( 1/8, i = /c = 0,
_ J 1/4, i > 0, A: = 0
- 1 (l/2)[(-l)'‘- - ll/(fc7r)^ i = 0,fc > 0,
I [(-1)* - ll/(fe7r)2, i>0.fc>0.
(35)
(36)
If the frequencies are closely spaced so uJn^<^ and hence Ly » uLy , then one positive
peak is expected once per cycle,
£;[iVo+]=a;/(27r)
(37)
886
and the probability density of narrow band peaks becomes,
N
t=0 fc=0
71=1
(38)
Figure 7 shows probability density of narrow band peaks for N=2,3, and 4 equal amplitude
(tti = 1) equal frequency series using equation (38). Each sum in equation (38) was carried
to 40 terms.
A power series solution for equation (20) can be found if all N terms in the series have
equal amplitude and frequency. The result is
Pa(A) =
Y2
1 1
4iV 4ivy2
A^
32Nr,i,
•h...
In the limit as N becomes infinite these equations become,
(39)
pYi-(y>y) =
IttY Y
Zr/i j rms*^ rms
(40)
Pa{A) =
y2
^ rms
(41)
Equation (40) is in agreement with an expression given by Crandall and Mark (1963, p.
47) and equation (41) is the Rayleigh distribution.
Equations (20), (35), (38), and (41) are conservative when applied to non-narrow band
processes in the sense that any troughs above the axis (points with y > 0 and dYjdt = 0
but d^Yldt^ > 0) are counted as peaks (Lin, 1967, p. 304; Powell, 1958; Broch, 1963).
Equations (35), (38) and (41) can provide probability distributions for peaks of narrow
band processes as a function of the number of sine waves from one to infinity and thus
they model random processes with peak-to-rms ratios from 2^/^ to infinity.
5. FATIGUE UNDER RANDOM LOADING
Fatigue tests are most often made with constant-amplitude sinusoidal loading. The
number of cycles to failure is plotted versus the stress that produced failure and the data
is often fitted with an empirical expression. MIL-HDBK-5G (1994) uses the following
empirical expression to fit fatigue data,
log iVy = + B2log{S^ - S4), = 5(1 - R)^K (42)
Here Nf is the cycles to failure during sinusoidal loading that has maximum stress 5 per
cycle. R is the ratio of maximum to minimum stress during a cycle. R = — 1 is fully
887
reversed stress cycling. Bi though B4 are fitted parameters. With a little work, we can
put this expression in the form used by Crandall and Mark (1963, p. 113).
JV = cSJ*- (43)
where Sd = 5(1 - - B4, c = 10-®' , and b = -82- For cycling in a time history that
has non constant amplitude, Miner-Palmgren proposed that the accumulated damage is
the sum of the ratios of the number of cycles at each amplitude to the allowing number of
cycles to failure at that amplitude (equations 42 and 43).
D = ^«(Si)/lV^(S,) (44)
i
where n{Si) is the number of cycle accumulated at stress amplitude Si and Nj is the
number of stress cycles at this amplitude which would cause failure.
Following Miles(1954) and Crandall and Mark(1963), the expected fractional damage
for a random stress cycling in system with dominant cycling at frequency f in time t/, is
= (45)
where Pa{S) is the probability density of a stress cycle having amplitude S and Nf{S) is
the number of allowable cycles to failure at this stress. Failure under random loading is
expected when the expected damage is unity. Setting = 1 at time such that
ftd = Nd, the inverse of the expected number of random vibration cycles to failure is
This expression can be used to create a fatigue curve for random cycling given the proba¬
bility density of the random stress cycle amplitudes (p^(5)) and a fatigue curve (equation
42 with parameters Bi though B4 and R) for sinusoidal cycling.
Substituting the probability density expression for narrow band amplitude (equation
35) and for the fatigue curve(equations 42 or 43) into equation (46) and integrating, we
obtain an expression for the expected number of cycles to failure as a function of the
number of sine waves and their amplitudes. For N equal amplitude sine waves this is,
<"->■■ - I ^1"
(47)
V (2 + 6)L,(l-B)®= '
+
iirBi
(1 - R)^Ly
mBi
■2’
.1 3
1 T-irr-^ f-** ^
(1 -
iV(B^(l-B)®’-B4)2,
ALlil - B)2B3 '
i^7r^(£„(l - B)®° - Bif.
4L|(1 - H)2®»
888
Recall that for this case Ly = Na, the rms value is Y^ms — {l/2)Na and the peak-
to-rms ratio is Peak/Yrms = y/2N (equations 2 though 4). is the generalized
hypergeometric function which is a series of polynomials. It is described by Gradshteyn,
Ryzhik, and Jeffrey (1994).
It is also possible to establish the fatigue curve using the Rayleigh distribution (equa¬
tion 18) and the MIL-HDBK-5 fatigue curve (equation 42). The result is
(2^'^r[l + 5]((1 -
-BlF,[l +
^ 3 Bl
2’2’2y;2„,(i-ij)2B3«
(48)
Bl
rr2Y,^^{i-RYB,^
_orf3 + il + 3 Bl
4 2’ 2’ 2V;2„,(1 - ii)2B3
iFi[..] is the confluent hypergeometric function which is described by Gradshteyn, Ryzhik,
and Jeffrey (1994).
Much of the complexity of these last two equations arises from the term B4 which is
associated with an endurance limit in the fatigue equation. That is, equation (42) predicts
that sinusoidal stress cycling with stress less than 54/(1— R) ^=3 produces no fatigue damage.
If we set B4 = 0 to set the endurance limit to zero, then equation (48) simplifies to.
This result for cycles to failure under Gaussian loading without an endurance limit is also
given by Crandall and Mark (1963, p. 117).
Equations (47), (48) and (49) allow us to compute the fatigue curves of a material
under random loading from a fatigue curve generated under sinusoidal loading (equation
43) for narrow band random processes with any peak-to-rms ratio from 2^/^ to infinity.
6. APPLICATION
Figure 8 is the MIL-HDBK-5G fatigue curve for aluminum 2024-T3 with a notch
factor of Kt=4 under sinusoidal loading with various R values. The fitted curve shown in
the figure, gives the following parameters for equation (42).
Bl = 8.3, B2 = -3.30, Bz = 0.66, B4 = 8.4
889
The B2 and B3 are dimensionless. B4 has the units of ksi, that is thousands of psi, and
10^^ has units of These Bi,..B4 are substituted into equations (43), (47), (48),
and (49).
The fatigue curves under random loading are computed as follows, 1) the number of
sine waves N is chosen and this fixes the peak-to-rms ratio from equation (3b), 2) set of
values of rms stresses are chosen and for each the corresponding sine waves amplitudes are
computed using equation (3b), a = Srmsy/VN (note that the peak stress much exceed
S4=8.5 ksi), and 3)the cycles to failure are calculated from equation (47) for finite peak-
to-rms ratios and equation (48) for Gaussian loading (infinite peak-to-rms).
For single sine wave, the peak-to-rms ratio is 2^/^, equation 4b, and the fatigue curve
interms of rms stress is adapted from the empirical data fit (equations 42, 43) by substi¬
tuting 2^^‘^SrTns for the stress amplitude.
Nd = c(2^/25.n..(l - - B^r^ (50)
where b= -B2 and c = 10-®^ Some results are shown in Figure 9 for R=-l.
7. CONCLUSIONS
Analysis has been made to determine the properties of a random process consisting of
the sum of a series of sine waves with deterministic amplitudes and random phase angles.
The joint probability density of the sum and its first two derivatives is determined. The
probability density of the sum and narrow band peaks have been found for an arbitrary
number of statistically independent sine wave terms. The fatigue cycles-to-failure resulting
from these processes has been found.
1. The peak-to-rms ratio of the sum of mutually independent terms exceeds unity. If ail
terms have the same peak and rms values then the peak-to-rms ratio of the series sum
increases with the square root of the number of terms in the series. The probability
of the series sum is zero beyond a maximum value, equal to the sum of the series
amplitudes, and below the minimum value. Hence, he probability densities of the
finite series, their peaks, and their envelope are non Gaussian.
3. The formulas allow the direct calculation of the probability density of the series and its
peaks from its power spectra density (PSD) under the assumption that each spectral
component is statistically independent.
4. The fatigue curves of a material under random loading with any peak-to-rms ratio
from 2^/^ to infinity can be computed dfrectly from the fatigue curve of the material
under sinusoidal loading.
890
REFERENCES
Abramowitz, M. and LA. Stegun 1964 Handbook of Mathematical Functions, National
Bureau of Standards, U.S. Government Printing OfRce, Washington D.C. Reprinted by
Dover.
Bennett, W.R., 1944 Acoustical Society of America 15, 165. Response of a Linear Rectifier
to Signal and Noise.
Bendat, J.S., 1958 Principles and Applications of Random Noise Theory, Wiley, N.Y.
Chandrasekhar, S., 1943,Reweiys of Modem Physics, 15, 2-74. Also available in Wax, N.
(ed) Selected Papers on Noise and Stochastic Processes, Dover, N.Y., 1954.
Cramer, H., 1970 Random Variables and Probability Distributions, Cambridge at the Uni¬
versity Press.
Crandall, S.H., and C. H. Mark 1963 Random Vibrations in Mechanical Systems, Academic
Press, N.Y.
Department of Defense, 1994 Metallic Materials and Elements for Aerospace Vehicle Struc¬
tures, MIL-HDBK-5G.
Gradshteyn, I.S., I.M. Ryzhik, and A. Jeffrey 1994 Table of Integrals, Series, and Products
5th Ed., Academic Press, Boston.
Lin, P.K., 1976 Probabilistic Theory of Structural Dynamics, Krieger, reprint of 1967 edi¬
tion with corrections.
Mathematica, 1995 Ver 2.2, Wolfram Research, Champaign, Illiinois.
Miles, J., 1954 Journal of Aeronautical Sciences 21, 753-762. On Structural Fatigue under
Random Loading.
Powell, A., 1958 Journal of the Acoustical Society of America SO No. 12, 1130-1135. On
the Fatigue Failure of Structure due to Vibrations Excited by Random Pressure Fields.
Rayleigh, J.W.S. 1880 Philosophical Magazine X 73-78. On the Resultant of a Large
Number of Vibrations of the Same Pitch and Arbitrary Phase. Also see Theory of Sound,
Vol 10, art. 42a, reprinted 1945 by Dover, N.Y.. and Scientific Papers, Dover, N.Y., 1964,
Vol. I, pp. 491-496.
Rice, S.O., 1944 The Bell System Technical Journal 23 282-332. Continued in 1945 24 ,
46-156. Mathematical Analysis of Random Noise. Also available in Wax, N. (ed) Selected
Papers on Noise and Stochastic Processes, Dover, N.Y., 1954.
Shmulei, U. and G.H. Weiss 1990 Journal of the American Statistical Association 85 6-19.
Probabilistic Methods in Crystal Structure Analysis.
Sveshnikov, A. A, 1968 Problems in Probability Theory, Mathematical Statistics and Theory
of Random Functions Dover, N.Y., translation of 1965 edition, pp. 74, 116.
891
Tolstov, G.P., 1962 Fourier Series, Dover, N.Y., pp. 173-177. Reprint of 1962 edition.
Weiss, G.H., 1994 Aspects and Applications of the Random Walk, North-Holland, Amster¬
dam.
Weiss, S.H. and U. Shmulei, 1987 Physica 146A 641-649. Joint Densities for Random
Walks in the Plane.
Willie, L.T., 1987 Physica 141 A 509-523. Joint Distribution Function for position and
Rotation angle in Plane Random Walks.
Wirsching, RH., T.L. Paez, and K. Ortiz 1995 Random Vibrations, Theory and Practice,
Wiley-Interscience, N.Y., pp. 162-166.
892
A
an
Bn
C{f)
cih.h)
E[N,]
E[No^]
iFi
pEq
f
i
3
Jo
k
K
Ly
Ey
m
N
Nf
n
Bviy)
Py(^)
pxYi^,y)
S
t
T
Y
Y
X
aij
r
lij
(f>n
U
a>n
NOMENCLATURE
amplitude, peak, or envelope
amplitude of the nth sine wave, a^, > 0
fitted parameter in equation (42)
characteristic function with parameter /
joint characteristic function with parameters fi and /2
expected number of positive peaks per unit time
expected number of zero crossing with positive slope per unit time
confluent hypergeometric function (Gradshteyn, Ryzhik, Jeffrey, 1994, art. 9.210)
generalized hypergeometric function (Gradshteyn, Ryzhik, Jeffrey, 1994, art.
9.210)
parameter in Fourier transform
integer index
imaginary constant,
Bessel function of first kind and zero order
integer index
complete elliptic integral of first kind, equation (33a)
+ ^2 + •• + sum of amplitudes
ujiai + uj2a2 + - • + sum of velocity amplitudes
integer index
number of terms in series
cycles to failure
integer index, n=l,2,..N
cumulative probability, the integral of Py{x) from x=— co to y
probability density of random parameter Y evaluated at T = a:
joint probability density of X and Y evaluated at Y = y and X = x
stress
time, 0 < t < T
length of time interval
sum of N modes or terms, —Ly<Y < Ly
first derivative with respect to time of Y, —Ly <Y< Ly
a random variable
dimensionless coefficient, equation (33)
gamma function, r[(2n + l)/2] = 7r^/^2“’^(2n - 1)!!
dimensionless coefficient, equation (36)
Dirac delta function
3.1415926..
XiX2.-xi\i, product of terms
phase angle of the nth sine wave, a uniformly distributed independent random
variable
circular frequency, a positive (non zero) real number
circular frequency of the nth term, a non zero integer multiple of 27r/T
893
Figure 1 Spectrum of vibration of a component on a turbojet engine cowling. Note the
finite number of distinct peaks.
894
Figure 2 Sample of the time history associated with the spectrum of Figure 1. Note the
signal is bounded, irregular and quasi sinusoidal.
895
-4.0 0 0.0 2.0
MO, OF S.D.
Figure 3 Probability density of the time history of Figures 1 and 2. Note that the maximum
values do not exceed plus or minus 2.5 standard deviations.
Yrms * (Probability Density of Y)
-3-2-10 1 2 3
Y / Yrms
- Normal Distribution
- Sine Wave Distribution
o Equation . N=10
• Equation , N=1
Figure 5 Normal probability density (equation 18) and sine wave probability density (equa¬
tion 6) in comparison with results of numerical integration of equation (13) for N=1 and
N=10.
898
Yrms* (Probability Density of Y)
Yrms (Probability Density of A)
- Two Sine Waves
- Three Sine Waves
. Four Sine Waves
Figure 7 Probability density of peaks in narrow band series with equal amplitudes (ai =
02.. = 1) and frequencies.
900
10*
in' {o' 10* 10* 10*
FRTIGUE LIFE. CYCLES
FIGURE 3.2.3.1.8(h). Besi-fit SI N curves for noiched. K, ~ 4.0 of 2024-T3 aluminum alloy sheet,
longitudinal direction.
Figure 8 Fatigue curves for notched 2024-T3 aluminum alloy with Kt=4. MI1-HDBK-5G
(1994, p. 3-115)
902
ACOUSTIC FATIGUE I
Strain Power Spectra of a Thermally Buckled Plate
in Random Vibration
Jon Lee and Ken R. Wentz
Wright Laboratory (FIB)
Wright-Patterson AFB, OH 45433, USA
Abstract
Several years ago, Ng and Wentz reported strain power spectra measured
at the mid-point of a buckled aluminum plate which is randomly excited by an
electrodynamic shaker attached to the clamped-plate boundary fixture. We
attempt to explain the peculiar features in strain power spectra by generating
the corresponding power spectra by the numerical simulation of a single-mode
equation of motion. This is possible because the essential dynamics takes place
in the frequency range just around and below the primary resonance frequency.
1. Introduction
For high performance military aircraft and future high-speed civil transport
planes, certain structural skin components are subjected to very large acoustic
loads in an elevated thermal environment [1]. This is because high-speed
flights call for a very powerful propulsion system and thereby engendering
acoustic loads in the anticipated range of 135-175 dB. More importantly,
because of the aerodynamic heating in hypersonic flights and the modern trend
in integrating propulsion sub-systems into the overall vehicular configuration,
some structural components must operate at high temperatures reaching up to
1300°F. Hence, the dual effect of thermal and acoustic loading has given rise
to the so-called thermal-acoustic structural fatigue [2,3].
Generally, raising the plate temperature uniformly but with an immovable
edge boundary constraint would result in thermal buckling, just as one observes
flexural buckling as the inplane stress along plate edges is increased beyond a
certain critical value. This equivalence has been recognized [4,5] and
exploited in previous analytical and experimental investigations of the thermal-
acoustic structural fatigue [6,7,8]. An experimental facility for thermal-
acoustic fatigue, termed the Thermal Acoustic Fatigue Apparatus, was
constructed at the NASA Langley Research Center in the late 80’ s. Under the
acoustic loading of 140-160 dB, Ng and Clevenson [9] obtained some strain
measurements of root-mean-square value and power spectral density (PSD) on
an aluminum plate heated up to 250^. Later, Ng and Wentz [10] have
repeated the heated Aluminum plate experiment but by randomly exciting the
clamped-plate boundary fixture by a shaker, and thereby recovering similar
strain measurements.
It should be noted that Ng and his colleagues [7,9,10] were the first to
achieve sufficient plate heating to induce thermal buckling and thus observe the
erratic snap-through under the acoustic or shaker excitations. Here, by erratic
903
we mean that a snap-through from one static buckled position to another takes
place in an unpredictable fashion. We reserve the adjective chaotic for a snap-
through occurring under the deterministic single-frequency forcing [1 1,12]. It
has already been observed that certain of the buckled plate experiment can be
explained, at least qualitatively, by a single-mode model of plate equations.
This is also validated by a theoretical analysis. Indeed, we showed that a single¬
mode Fokker-Planck formulation can predict the high-temperature moment
behavior and displacement and strain histograms of thermally buckled plates,
metallic and composite [13,14].
In retrospect, a single-mode model has proven more useful than originally
intended. That is, the single-mode Fokker-Planck formulation of an isotropic
plate lends itself to predicting certain statistics of composite plates which are
simulated by multimode equations or tested experimentally by multimode
excitations. For a refined and more quantitative comparison, one must inject
more realism into dynamical models by including the multimode interactions.
However, before giving up the single-mode plate equation, there is an
important problem that this simple model is well suited for investigation. That
is, prediction of the strain PSD measurement by Ng and Wentz [10]. As we
shall see in Sec. 4, the strain PSD of a thermally buckled plate exhibits a strong
spectral energy transfer toward zero frequency, and thereby saturating
frequency range well below the primary resonance frequency. This downward
spectral energy transfer can be modeled quite adequately by the single-mode
plate equation without necessitating multimode interactions.
2. Equation of motion for the aluminum plate experiment
By the Galerkin procedure, the von Karman-Chu-Herrmann type of large-
deflection plate equations give rise to infinitely coupled modal equations [15].
However, much has been learned from a prototype single-mode equation for
displacement^ [13,14].
q + Pq + k„{l-s)q + aq^ = g„ + g{t), (1)
where the overhead dot denotes d/dt and the viscous damping coefficient is
P = 2^^ with damping ratio ^ . For the clamped plate, we have
;i„=f(r‘'+2rV3 + l),
s = rji + (1 -M) (1 + (r^+ ir^) /6] ,
a = ^{(7^+r'^+2^i) + |(i-/i^)[T(r^+r'^) +^(.r+r''T^
+ (r+47'‘)‘^ + (47+ 7"‘r^]} ,
&= (r‘'+2r^/3 +i)Sjj6.
Note that the expressions for s and g„ are specific to the typical temperature
904
variation and gradient profiles assumed in Ref. [15]. Here, 7 = b/a is the
aspect ratio of plate sides a and b, and fi is Poisson's ratio. The uniform plate
temperature is measured in units of the critical buckling temperature. The
maximum temperature variation on the mid-plate plane is denoted by and
TJ5g is the maximum magnitude of temperature gradient across the plate
thickness, where 5^ and 5 ^ are scale factors. Hence, 0 signifies no
temperature variation over the mid-plate plane, and 0 zero temperature
gradient across the plate thickness. Finally, g{t) denotes the external forcing.
The parameter s represents thermal expansion due to both the uniform
plate temperature rise above room temperature and temperature variation over
the mid-plate uniform temperature. The combined stiffness k^(l - s)q consists
of the structural stiffness k^q and thermal stiffness -sk^q , which cancel each
other due to the sign difference. It is positive for 5 <1, then Eq. (1) has the
form of Duffing oscillator with a cubic term multiplied by a , which represents
geometric nonlinearity of membrane stretching. For s >1 Eq. (1) reduces to
the so-called buckled-beam equation of Holmes [11] with a negative combined
stiffness. In contrast, denotes thermal moment induced by a temperature
gradient across the plate thickness; hence, it appears in the right-hand side of
Eq. (1) as an additional forcing. The interplay of the terms involving 5, a, and
g^ can best be illustrated by the potential energy [15]
U{q) = -go q + k^(X-s) (fn -H a . (2)
Fig. 1 shows that V{c^ is symmetric when g„ = 0. For s<l it has a single well
which splits into a double well as s exceeds unity. Note that the distance
between the twin wells increases as for large s (Fig. 1(b)). This
interpretation is valid approximately for go>^- That is, a positive g^ lowers
the positive side potential (^>0) and raises the negative side potential {q<0),
and thereby rendering the potential energy asymmetric.
U(q) ^(^1)
Fig. 1 Potential energy, (a) s<V, (b) ^ >1, w = ^k^(s - l)/a , d = - l))V4a.
( - ^.=0; --- 5„>0)
905
It must be pointed out that Eq. (1) is dimensionless and involves explicitly
only 7 and /i. For the aluminum plate experiment [10], 7=10 in./8 in. and
so that k^=Q3.9l and a =85. 33. If we further assume 5^- 0 for
simplicity, the thermal parameter reduces to s=T^. Previously, Eq. (1) was
used for the investigation of stationary Fokker-Planck distribution which
involves only the ratio p!F, where F is the constant power input [13,14].
Hence, nondimensionalization has indeed spared us from specifying in detail
other plate parameters. Things are however different in numerical simulation
because we must know the characteristic scales to correctly interpret time-
dependent solutions. By retracing the derivation, we find that the dimensionless
quantities in Eq. (1) are (Eq. (IV. 1) in Ref. [1])
q-qlh, t-t/t*, g = glg*, (3)
where the overhead bar denotes the physical quantity. Here, the plate thickness
h, t*={b/Kf.y[ph/D, and g^=p(h/t*)^ are the characteristic length, time, and
force, respectively (p = mass density, D=Eh^f\2{\~p}) , £= Young’s
modulus of elasticity), as listed in Table L We now rewrite Eq. (1) with the
numerical coefficients (Table I). _
q + 0.0978? + 23.91(1 -s)q + 85.33?^ = (4)
where g{t) has the unit of psi.
Table I. Parameter values for the aluminum plate experiment
7, «
10in./8in., VoX 0.01, 23.910, 85.332
5.. 5,
p
o
h, f*, g'^-
0.05 in., 3.305 lO'^sec.^, 5.806 1 O'’ psi^
(+) p = 0.098 Ib/in^ and E =1.03 10’ psi.
3. Monte-Carlo simulation
Because of 5 =0, Eq. (4) has the standard form of Duffing (s<l) and
Holmes (s>l) oscillators. In stead of a single frequency for forcing g{t) [11,
12], in Monte-Carlo simulation all forcing frequencies are introduced up to a
preassigned maximum so that forcing represents a plausible physical
realization. Of course, particular interest here is a constant PSD. We shall
begin with generation of a time-series for random processes with such a PSD.
3.1 Random forcing time-series
We adopt here the procedure for generating a time-series of Shinozuka
and Jan [16], which has been used for a oscillator study [17] and extensively
for structural simulation applications by Vaicaitis [18,19]. Since it relies
heavily on the discrete fast Fourier transforms, such as FFTCF and FFTCB
subroutines of the IMSL library, it is more expedient to describe the procedure
operationally rather than by presenting somewhat terse formulas. Let us
906
introduce Nj: frequency coordinates which are equally spaced in
the band width A/=/^3x/A^^. Now, the task is to generate a time-series of total
time T that can resolve up to . Assume T is also divided into time
coordinates with the equal time interval At=T/Nj, From the time-frequency
relation r=l/A/, we find ^At. If we choose
N
Nf = (5)
is the Nyquist frequency, consistent with our original definition of the
upper frequency limit of resolution.
A random time-series with a constant PSD can be generated in the
following roundabout way. We begin by assuming that we already have a
forcing power spectrum ^g{f) of constant magnitude over [0,/njax]- Such a
PSD may be represented by a complex array A„= VC exp(~27rz0„) (n = 1,
Nf), where takes a random value distributed uniformly in [0, 1]. Clearly the
magnitudes of are C, hence We then enlarge the complex array
A„ by padding with zeros for n =Ny+l, and Fourier transform it to
obtain a complex array B^{n= N^). The random time-series for is
now given by the real part of
= Real part of (n = 1, N^) (6)
As it turns out, when g „ is padded with zeros for the imaginary components
and Fourier transformed, we recover the original array A„ (w=l, Nf) with
Since the spectrum area is nothing but total forcing power
<g^> (say, in psi^), we can relate C with the variance <g^> of pressure
fluctuations, which is often expressed by the sound pressure level (SPL) in dB,
according to SPL=10 log<g2 >/p2, where p=2,9 10*^ psi. Hence,
c = - . (7)
/max
Here, Eqs. (6) and (7) defined heuristically are meant to explain the
corresponding formulas (2) and (12) in Ref [18].
For the numerical simulation we first note that the resonance frequency of
Eq. (1) is /^=-y^/27r« 0.778 for s ~ 0. This gives the dimensional resonance
frequency fjt*~235.5 Hz which is somewhat larger than the experimental
217.7 Hz (Fig. 3(a)). As shown in Table II, we assign (~9/r) because
the electrodynamic shaker used in the experiment [10] has the upper frequency
limit 2000 Hz.
Table II. Dimensionless parameter values for the numerical simulation
at, N, _ 7, 8192, 4096
At, T 0.071. 585
907
3.2 Displacement power spectrum
Under a random time integration of Eq. (4) yields a time-series for q^.
We first comment on the time integration. Although there are special solvers
[17,20] proposed for stochastic ordinary differential equations (ODEs), we
shall use here the Adams-Bashforth-Moulton scheme of Shampine and Gordon
[21], which has been implemented in DEABM subroutine of the SLATEK
library. Although DEABM has been developed for nonstochastic ODEs, its
use for the present stochastic problem may be justified in part by that one
recovers linearized frequency response functions by the numerical simulation
(Sec. 3.4). Obviously, this does not say anything about the strongly nonlinear
problem in hand, and it should be addressed as a separate issue. In any event,
DEABM requires the absolute and relative error tolerances, both of which are
set at no larger than 10“^ under the single-precision algorithm for time
integration. Note that actual integration time steps are chosen by the
subroutine itself, commensurate with the error tolerances requested. Recall
that is updated at every time interval Ar, and we linearly interpolate the
forcing value within A? .
We begin time integration of Eq. (4) from the initial configuration at the
bottom of the single-well potential, ^(0)=p(0)=0, for 5' <1 and the positive
side double-well potential, q{0)=^kj,s~\)/a andp(0)=0, for j >1. And we
continue the integration up to T. By Fourier transforming time-series q^, we
obtain displacement power spectrum 0^(/). This process of integrating and
transforming is repeated over three contiguous time ranges of 7, and the
successive PSDs are compared for stationarity. Since it is roughly stationary
after three repetitions, we report here only the PSD of the third repetition.
From the stationary input-output relation [22] where
is the magnitude of system frequency response function, we write
= (S)
Since O (/)=C, the and would have a similar functional
s
dependence upon /, Hence, we call them both the displacement PSD.
3.3 Strain power spectrum
Although displacement is the direct output of numerical simulation, one
measures strain rather than the displacement in plate experiment. At the
present level of plate equation formulation, the strain e is given
by the quadratic relation
e = + C^q + C2(f‘ ■> (9)
where C,- are given at the middle {x/a = y/b =1/2) of a clamped plate as follows
(Appendix D of Ref [13])
908
c =
(/+2yV3+l)?;g,
3(1+At)(y"+1)
{l-li}-
q =
8y^
3 ’
r 32 fy^ 5n (l-/iyV4)
^ 9 [2 16 2(y+y-‘f (y+4y-‘)^ (4y + y‘fJ
For we have C^=0, Q =4.17, and Q =2.77 (Table I). Hence, Eq. (9)
engenders only the linear and quadratic transformations, but no translation. In
any event, translation has no effect on the spectral energy contents. By Fourier
transforming time-series (n=l, A^^), we obtain strain power spectrum
. Although the forcing PSD is not constant, one computes the forcing
spectrum ratio as in Eq. (8) and call it the magnitude square of strain frequency
response function for the lack of a better terminology.
3.4 The linear oscillators
For the pre-buckled (5 <1) linear oscillator (a= 0) we rewrite Eq. (4) in
standard form _
q + + 0)1(1 -s)q = (10)
where col=k^, and obtain
I H^(f)^ = [(0)1(1 -s)- + (An^co^ffT'- (11)
As shown in Fig. 2(a), the numerical simulation of Eq. (10) recovers
as given by Eq. (11) over the entire frequency range. Although the simulation
of Fig. 2(a) was carried out with SPL=130 dB, it does not depend on SPL
since Eq. (10) is linear. Physically speaking, Eq. (10) oscillates in a single-well
potential (Fig. 1(a)). Since the potential energy has two wells (Fig. 1(b)) for
s >1, we linearize Eq. (1) around the positive side potential well by the
transformation q=q'+^k^{s-l)la . Hence, the corresponding linear oscillator
is
(a) (b)
Fig. 2 Linear frequency response functions, (a) Displacement; (b) Strain
(j = 0; - Numerical simulation; • Eq. (U))
909
q' + 2^0)^ q’ + Icolis -V)q' = (12)
s
In parallel to Eq. (1 1), the frequency response function of a post-buckled plate
I = [(^-colis - 1) - 4;rV^)2 + . (13)
The resonance frequency f=co^^2{s-l)/27J: of a post-buckled (s >1) plate
should be compared with f=co^^2{\~s)/2n of the pre-buckled (j <1) plate.
Now, for the linear oscillators we see that is also given by Eq. (1 1)
and (13) for ^ <1 and >1, respectively (Fig. 2(b)). This is because the spectral
energy distribution is not at ail affected by a linear transformation.
4. Displacement and strain power spectra
As we shall see in Sec 4.1, the experimental strain PSD exhibits downward
spectral energy transfer toward zero frequency, so that there is a considerable
spectral energy buildup below the resonance frequence as SPL is raised.
Moreover, it also involves an upward spectral energy transfer which then
contributes to both the increased resonance frequency and broadened
resonance frequency peak. Since spectral energy transfers take place around
and below the primary resonance frequency, it is possible to depict the
downward and upward spectral energy transfers by the numerical simulation of
Eq. (4) without necessitating multimode interactions. We shall first discuss the
characteristic features of experimental strain PSDs.
4. 1 Experimental strain PSD
Of the spectra reported in Ref. [10], we consider the following two sets.
One is the nonthermal set (^=0) consisting of two PSDs of small and large
SPLs. The other is the post-buckled set (5=1.7) of four PSDs. For the
convenience of readers, we have reproduced in Figs. 3 and 4 the selected PSDs
from Ref. [10] by limiting the upper frequency to 600 Hz, and the pertinent
data are summarized in Table m.
Table in. Strain power spectra of experiment and numerical simulation
Fig. 4(a) Fig. 6a
Fig. 4(b) Fig. 6b
Fig. 4(c) Fig. 6c
Fig. 4(d) Fig. 6d (*)
Fig. 7
Fig. 8
Fig. 9
Fig. 10
(*) Computed from the acceleration a measured in units of g.
910
Fig. 4 Experimental strain PSD =1.7). (a) 130.1dB; (b) 142dB; (c) 151.5dB; (d) 154.6dB
The following observations are drawn from the experimental PSDs. First,
for the nonthermal plate
Figure 3 fa): Compare the measured strain fr-2\l Hz with the theoretical
displacement 235 Hz of Eq. (4). Note that a small spectral energy peak is
found at 467 Hz which is about twice (-2.15) the strain value.
911
Figure Sfb): With SPL~150 dB the strain increases to 240 Hz and the
spectral width at the half resonance peak has nearly doubled. The spectral
energy buildups at zero and 515 Hz are more noticeable than in Fig. 3(a).
Again, 515 Hz is about twice (-2.15) the primary strain f,.
(a) (b)
f f
Fig. 5 Numerical simulation results under .y=0 and SPL=130 dB.
(a) Displacement ( - simulation,* Eq. (11)); (b) Strain ( - simulation, • Eq. (11));
(c) PSD averaged over 12 frequency intervals ( - displacement, — • — strain);
(d) Strain PSD.
Next, for the thermally buckled plate
Figure 4(a): The primary strain fr=227 Hz should be compared with the
theoretical displacement /^=279 Hz of Eq. (13). A second spectral energy
peak is found at 537 Hz, much larger than twice (-2.37) the primary strain /^.
Figure 4rb): Here, the spectral energy buildup is most significant at zero
frequency. Besides, there appear two spectral energy humps at 100 and 183
Hz, below the primary strain = 227 Hz of Fig. 4(a). Discounting the zero-
frequency spectral peak, PSD may be approximated by a straight line in the
semi-log plot, hence it is of an exponential form up to 400 Hz.
Figure 4rc): The zero-frequency peak is followed by a single spectral energy
hump at 115 Hz. Again, PSD can be approximated by a straight line and its
slope is roughly the same as in Fig. 4(b).
Figure 4rd): A major spectral energy peak emerges at 130 Hz, followed by a
minor one at 350 Hz. Theoverall spectral energy level is raised so that the
magnitude of PSD ranges over only two decades in the figure.
In Figs. 4(b)-(d) we have ignored the spectral energy peaks at around 500
Hz, for they are not related to the first plate mode under consideration. This is
further supported by the simulation evidence to be discussed presently.
4.2 Numerical simulation results
After choosing .y = 0 or 1.7, we
are left with SPL yet to be specified. o
Ideally, one would like to carry out the
numerical simulation of Eq. (4) by
using SPL of the plate experiment 'm-z
(Table III) and thus generate strain ^
PSDs which are in agreement with
Figs. 3 and 4. Not surprisingly, the _4
reality is less than ideal. An obvious
reason that this cannot be done is that
the forcing energy input is fed into all f
plate modes being excited in Hg. 6 PSD averaged over 12 frequency
experiment, whereas the forcing (j=0, SPL=138dB)
energy excites only one mode in the - displacement; -•-strain
numerical simulation. Consequently, SPL for the numerical simulation should
be less than the experimental SPL, but we do not know a priori how much
less. We therefore choose a SPL to bring about qualitative agreements
between the single-mode simulation and multimode experiment. As anticipated,
the simulation SPLs (Table HI) are consistently smaller than the experimental
values.
The numerical simulation results are shown in Figs. 5-6 for 5 = 0 and Figs.
7-10 for s =1.7. Actually each figure has four frames, denoted by (a)-(d).
First, frames (a) and (b) depict and Since they are very
jagged at large SPLs, we average the spectral energy over 12 frequency
intervals and present both of the smoothed-out frequency response functions in
the same frame (c). Lastly, frame (d) shows Og(/) itself Since there is no
qualitative difference between <E>g(/) and we shall call them both the
strain PSD. We present all four frames (a)— (d) of Figs. 5 and 7, but only the
frame (c) of Figs. 6, 8, 9 and 10 here for the lack of space.
First, for the nonthermal plate
Figure 5: The simulated is closely approximated by Eq. (11) with f =
236 Hz. Note that is also approximated by Eq. (11) for all frequencies
913
Fig. 7 Numerical simulation results under j=1.7 and SPL=129 dB.
(a) Displacement ( - simulation,* Eq. (13)); (b) Strain ( - simulation, • Eq. (13));
(c) PSD averaged over 12 frequency intervals ( - displacement, — • strain);
(d) Strain PSD.
but zero and 476 Hz, where the strain spectral energy piles up due to the
quadratic transformation (9). Since 476 Hz is nearly twice (-2.02) the primary
/^, strain spectral energy buildups are due to the sum and difference of two
nearly equal frequencies, ± /2, where/i==/2^/^.
Figure 6: The primary strain is shifted slightly upward to 253 Hz and the
spectral width at half resonance peak is 50% wider than that of Fig. 5(c). The
spectral energy builds up at 525 Hz which is roughly twice (-2.08) the /^. At
SPL=138 dB we find that the strain spectral energy hump at 525 Hz is about 2
decades below the resonance frequency peak, as was in Fig. 3(b).
Now, for the thermally buckled plate
Figure 7: The simulated and are weU approximated by Eq.
(13) around /^=270 Hz which is a litde below the linearized /^=279 Hz.
Unlike in Fig. 5 for 5=0, both and l/7^(/)F show spectral energy
building up significantly near zero and 543 Hz which is twice (-2.01) the /^.
914
2p — ! - 1 - ! - 1 - 1 - T— n - 1 - r
^ _ 1 - 1 - 1 - < - 1 - > - 1 - ^ - 1
0 300 600
f
Fig. 8 PSD averaged over 12 frequency
intervals (j =1.7, SPL=138 dB)
- displacement; — • — strain
Note that in Fig. 7(a) the spectral
energy hump at 543 Hz is about 3
decades below the primary frequency
peak, as was in Fig. 4(a).
Figure 8: After a large zero-frequency
peak, two spectral energy humps
appear at 131 Hz and 236 Hz. Note
that the ratios of these frequencies to
the /, (131/279 -0.47 and 236/279 -
0.85) are comparable with the same
ratios (100/227 -0.44 and 183/227 -
0.81) found in Fig. 4(b). Excluding
the zero-frequency peak, the overall
strain PSD is a straight line, hence of
an exponential form, as in Fig. 4(b),
Figure 9: The zero-frequency spectral peak is followed by a single major
energy hump at 154 Hz. The ratio of this to the (154/279 -0.56) is
somewhat larger than the ratio (115/227 -0.51) in Fig. 4(c). The strain PSD
can also be approximated by a straight line over the entire frequency range, and
Figs. 8 and 9 seem to have the same slope when fitted by straight lines.
Figure 10: The spectral magnitude of is larger than that of in
the frequency range above 300 Hz. The choice of SPL=146 dB was based on
that the PSD magnitude around 300 Hz is about 2 decades below the main
spectral peak magnitude at 180 Hz, thus emulating Fig. 4(d).
All in all, by numerical simulations we have successfully reproduced the
peculiar features in the two sets of strain PSDs observed experimentally under 5
= 0 and 1.7.
Fig. 9 PSD averaged over 12 frequency
intervals (s =1.7, SPL=143 dB)
- displacement; — • — strain
Fig. 10 PSD averaged over 12 frequency
intervals (j =1.7, SPL=146 dB)
- displacement; — • — strain
915
5. Concluding remarks
At low SPL the nonthermal {s= 0) and post-buckled (^=1.7) plates appear
to have a similar PSD. However, this appearance is quite deceptive in that the
nonthermal plate motion is in a single-well potential, so that PSD does not
change qualitatively as SPL is raised. On the other hand, the trajectory of a
post-buckled plate is in one of the two potential energy wells when SPL is very
small. However, as we raise SPL such a plate motion can no longer be
contained in a potential well, and hence it encircles either one or both of the
potential wells in an erratic manner. This is why the experimentally observed
and numerically simulated strain PSDs of a post-buckled plate exhibit
qualitative changes with the increasing SPL, and thereby reflect the erratic
snap-through plate motion. A quantitative analysis of snap-through dynamics
will be presented elsewhere.
Lastly, we wish to point out that a PSD of straigh-line form in the semi-log
plot was observed in a Holmes oscillator when trajectories are superposed
randomly near the figure-eight separatrix [23].
Acknowledgments
Correspondence and conversations with Chung Fi Ng, Chuh Mei, Rimas
Vaicaitis, and Jay Robinson are sincerely appreciated. We also wish to thank
the referees for their helpful suggestions to improve the readability of this
paper.
References
1. Lee, J., Large-Amplitude Plate Vibration in an Elevated Thermal
Environment, WL-TR-92-3049, Wright Lab., Wright-Patterson AFB, OH,
June, 1992.
2. Jacobson, M.J. and Maurer, O.F., Oil canning of metallic panels in
thermal-acoustic environment, AIAA Paper 74-982, Aug., 1974.
3. Jacobson, M.J., Sonic fatigue of advanced composite panels in thermal
environments, J. Aircraft, 1983, 20, 282-288.
4. Bisplinghoff, R.L. and Pian, T.H.H., On the vibrations of thermally
buckled bars and plates, in Proc. 9th Inter. Congr. of Appl. Mech.,
Brussels, 1957, 7, 307-318.
5. Tseng, W.-Y., Nonlinear vibration of straight and buckled beams under
harmonic excitation, AFOSR 69-2157TR, Air Force Office of Scientific
Research, Arlington, VA, Nov., 1969.
6. Seide, P. and Adami, C., Dynamic stability of beams in a combined
thermal-acoustic environment, AFWAL-TR-83-3072, Flight Dynamics
Lab., Wright-Patterson AFB, OH, Oct., 1983.
7. Ng, C.F., Nonlinear and snap-through responses of curved panels to
intense acoustic excitation, /. Aircraft, 1989, 26, 281-288.
8. Robinson, J.H. and Brown, S.A., Chaotic structural acoustic response of a
milled aluminum panel, 36th Structures, Structural Dynamics, and
916
Material Conference, AIAA-95-1301-CP, New Orleans, LA, 1240-1250,
Apr. 10-13, 1995.
9. Ng, C.F. and Clevenson, S. A., High-intensity acoustic tests of a thermally
stressed plate, J, Aircraft, 1991, 28, 275-281..
10. Ng, C.F. and Wentz, K.R., The prediction and measurement of thermo¬
acoustic response of plate structures, 31st Structures, Structural
Dynamics, and Material Conference, AIAA-90-0988-CP, Long Beach,
CA, 1832-1838, Apr. 2-4, 1990.
11. Holmes, P., A nonlinear oscillator with a strange attractor, Phil. Trans.,
Roy. Soc. of London, 1979, 292A, 419-448.
12. Dowell, E.H. and Pezeski, C., On the understanding of chaos in Duffings
equation including a comparison with experiment, J. Appl. Mech., 1986,
53, 5-9.
13. Lee, J., Random vibration of thermally buckled plates: I Zero temperature
gradient across the plate thickness, in Progress in Aeronautics and
Astronautics, 168, Aerospace Thermal Structures and Materials for a
New Era, Ed. E.A. Thornton, AIAA, Washington, DC, 1995. 41-67.
14. Lee, J., Random vibration of thermally buckled plates: n Nonzero
temperature gradient across the plate thickness, to appear in J. Vib. and
Control, 1997.
15. Lee, J., Large-amplitude plate vibration in an elevated thermal
environment, Mech. Rev., 1993, 46, S242-254.
16. Shinozuka, M. and Jan, C.-M., Digital simulation of random processes
and its applications, J. Sound and Vib. 1912, 25, 1 1 1-128.
17. Chiu, H.M. and Hsu, C.S., A cell mapping method for nonlinear
deterministic and stochastic systems - Part II: Examples of application, J.
Appl. Mech., 1986, 53, 702-710.
18. Vaicaitis, R., Nonlinear response and sonic fatigue of national aerospace
space plane surface panels, J. Aircraft, 1994, 31, 10-18.
19. Vaicaitis, R., Response of Composite Panels Under Severe Thermo-
Acoustic Loads, Report TR-94-05, Aerospace Structures Information and
Analysis Center, Wright-Patterson AFB, OH, Feb., 1994.
20. Kasdin, N.J., Runge-Kutta algorithms for the numerical integration of
stochastic differential equations, J. Guidance, Control, and Dynamics,
1995,18, 114-120.
21. Shampine, L.F. and Gordon, M.K., Computer solution of ordinary
differential equations, 1975, Freeman, San Francisco.
22. Lin, Y.K., Probabilistic theory of structural dynamics, Robert E. Krieger
Publishing, 1976, Huntington, NY.
23. Brunsden, V., Cortell, J. and Holmes, P.J., Power spectra of chaotic
vibrations of a buckled beam, J. Sound and Vib., 1989, 130, 1-25.
917
918
ENHANCED CAPABELITIES OF THE NASA LANGLEY
THERMAL ACOUSTIC FATIGUE APPARATUS
Stephen A. Rizzi and Travis L. Turner
Structural Acoustics Branch
NASA Langley Research Center
Hampton, VA 23681-0001
ABSTRACT
This paper presents newly enhanced acoustic capabilities of the Thermal
Acoustic Fatigue Apparatus at the NASA Langley Research Center. The
facility is a progressive wave tube used for sonic fatigue testing of aerospace
structures. Acoustic measurements for each of the six facility configurations
are shown and comparisons with projected performance are made.
INTRODUCTION
The design of supersonic and hypersonic vehicle stmctures presents a
significant challenge to the airframe analyst because of the wide variety and
severity of environmental conditions. One of the more demanding of these is
the high intensity noise produced by the propulsion system and turbulent
boundary layer [1]. Complicating effects include aero-thermal loads due to
boundary layer and local shock interactions, static mechanical preloads, and
panel flutter. Because of the difficulty in accurately predicting the dynamic
response and fatigue of structures subject to these conditions, experimental
testing is often the only means of design validation. One of the more common
means of simulating the thermal- vibro-acoustic environment is through the use
of a progressive wave tube. The progressive wave tube facility at NASA
Langley Research Center, known as the Thermal Acoustic Fatigue Apparatus
(TAFA), has been used in the past to support development of the thermal
protection system for the Space Shuttle and National Aerospace Plane [2]. It
is presently being used for sonic fatigue studies of the wing strake
subcomponents on the High Speed Civil Transport [3].
The capabilities of the TAFA were previously documented by Clevenson and
Daniels [4]. The system was driven by two Wyle WAS 3000 airstream
modulators which provided an overall sound pressure level range of between
125 and 165 dB and a useful frequency range of 50-200 Hz. A 360 kW quartz
lamp bank provided radiant heat with a peak heat flux of 54 W/cm^. A
schematic of the facility is shown in Figure 1. Representative spectra and
coherence plots are shown in Figures 2 and 3. Since that time, the facility has
undergone significant enhancements designed to improve its acoustic
capabilities; the heating capabilities were not changed. The objectives of the
enhancements were to increase the maximum overall sound pressure level
(OASPL) to 178 dB, increase the frequency bandwidth to 500 Hz and improve
the uniformity of the sound pressure field in the test section. This paper
919
documents the new capabilities of the TAFA and makes comparisons with the
projected performance.
Figure 1: Schematic of the old TAFA facility.
Figure 2: Test section spectra of the Figure 3: Test section coherence of the
old TAFA facility. old TAFA facility.
FACILITY DESCRIPTION
In order to meet the design objectives, extensive modifications were made to
the sound generation system and to the wave tube itself. A theoretical increase
of 6 dB OASPL was projected by designing the system to utilize eight WAS
3000 air modulators compared to the two used in the previous system. A
further increase of nearly 5 dB was expected by designing the test section to
accommodate removable water-cooled insert channels which reduced its cross-
sectional area from 1.9m x 0.33m to 0.66m x 0.33m. The frequency range was
increased through the use of a longer horn design with a lower (15 Hz vs. 27
Hz in the old facility) cut-off frequency, use of insert channels in the test
section to shift the frequency of significant standing waves above 500 Hz, and
design of facility sidewall stmctures with resonances above 1000 Hz. The
uniformity of the sound pressure field in the test section was improved through
several means. A new, smooth exponential horn was designed to avoid the
impedance mismatches of the old design. To minimize the effect of
uncorrelated, broadband noise (which develops as a byproduct of the sound
920
generation system), a unique design was adopted which allows for the use of
either two-, four-, or eight-modulators. When testing at the lower excitation
levels for example, a two-modulator configuration might be used to achieve a
lower background level over that of the four- or eight-modulator
configurations. In doing so, the dynamic range is extended. Lastly, a catenoidal
design for the termination section was used to smoothly expand from the test
section.
Schematics of the facility in the three full test section configurations are shown
in Figures 4-6. In the two-modulator configuration, the 2 x 4 transition cart
acts to block all but two of the eight modulators. The facility is converted
from the two- to four-modulator configuration by the removal of the 2 x 4
transition cart and connection of two additional modulators. In doing so, the
modulator transition cart slides forward and thereby maintains the continuous
exponential expansion of the duct. In the four-modulator configuration, the 4
X 8 transition cart acts to block the two upper and two lower modulators.
Removal of this component and connection of the four additional modulators
converts the facility to the eight-modulator configuration. Again, the
continuous exponential expansion is maintained as the modulator transition
cart slides forward.
Figure 4: Two-modulator full test section configuration.
921
Figure 5: Four-modulator full test section configuration.
Figure 6: Eight-modulator full test section configuration.
Schematics of the three reduced test section configurations are shown in
Figures 7-9. In these configurations, the horn cart is discarded and the horn
transition cart mates directly to the test section. Water-cooled inserts are used
in the test section to reduce its cross-sectional area. Upper and lower inserts in
the termination section are used to smoothly transition the duct area to the full
dimension at the exit. Conversion from the two- to the four-modular
configuration and from the four- to the eight-modulator configuration is again
accomplished through removal of the 2 x 4 and 4x8 transition carts,
respectively.
922
HORN TTUNSTTION SECTION
Figure 7: Two-modulator reduced test section configuration.
Figure 8: Four-modulator reduced test section configuration.
TEST PROCEDURE
Measurements were taken for several conditions in each of the six facility
configurations. Each modulator was supplied with air at a pressure of 207 kPa
(mass flow rate of approximately 8.4 kg/s) and was electrically driven with the
same broadband (40-500 Hz) signal. Acoustic pressures were measured at
several locations along the length of the progressive wave tube using B&K
model 4136 microphones and Kulite model MIC-190-HT pressure transducers,
see Table 1. The positive x-direction is defined in the two-modulator full
configuration (from the modulator exit) along the direction of the duct. The
positive y-direction is taken vertically from the horizontal centerline of the
923
HORN TRIWSmOH SECTION
MODULATOR TRANSITION
FLEMSie HOSE
adapter puts assembly
TEST SECTION
ADAPTER plate assembly
TERM1ASAT10N SECTION
horn TRANSITION CART
Figure 9: Eight-modulator reduced test section configuration.
duct and the positive z-direction is defined from the left sidewall of the duct as
one looks downstream.
Table 1: Kulite (K) and microphone (M) locations of acoustic measurements.
Loc.
Description
Type
Coordinate (m)
1
Test Sect. Horizontal Centerline Upstream
K
7.75, 0, 0
2
Test Sect. Horizontal Centerline Downstream
K
8.71, 0, 0
5
Test Sect. Vertical Centerline Top
M
8,23, 0.3, 0
15
Test Sect. HorizontaWertical Centerline
K
8,23, 0, 0
25
Test Sect. Vertical Centerline Bottom
M
8.23, -0.3, 0
28
2x4 HorizontaWertical Centerline
M
2.19, 0,0
29
4x8 Horizontal Centerline, % Downstream
M
3.66, 0, 0
30
Horn Tran, Hor. Centerline, % Downstream
M
4.75, 0, 0
35
Termination HorizontaWertical Centerline
M
12.46, 0, 0.17
The acoustic pressure at location 1 was used as a reference measurement for
shaping the input spectrum and for establishing the nominal overall sound
pressure level for each test condition. For each configuration, the input
spectrum to the air modulators was manually shaped through frequency
equalization to produce a nearly flat spectrum at the reference pressure
transducer. Data was acquired at the noise floor level (flow noise only) and at
overall levels above the noise floor in 6 dB increments (as measured at the
reference location) up to the maximum achievable. Thirty-two seconds of
time data were collected at a sampling rate of 4096 samples/s for each
transducer in each test condition. Post-processing of the time data was
performed to generate averaged spectra and coherence functions with a 1-Hz
frequency resolution.
924
RESULTS
For each facility configuration, plots of the following quantities are presented:
normalized input spectrum to the air modulators, minimum to maximum
sound pressure levels at the reference location, maximum sound pressure
levels in the test section, maximum sound pressure levels upstream and
downstream of the test section, and vertical and horizontal coherence in the
test section. The minimum levels in each case correspond to the background
noise produced by the airflow through the modulators.
Normalized input voltage spectra to each modulator for each configuration are
shown in Figures 10, 15, 20, 25 and 30. These spectra were generated to
achieve as flat an output spectrum as possible at the reference location for the
frequency range of interest (40-200 Hz for the full section, 40-500 Hz for the
reduced section). As expected, the significant difference between the full and
reduced configurations is seen in the high (>200 Hz) frequency content.
Figure 11 shows a background noise level of 126 dB (the lowest of all
configurations) for the two-modulator full test section configuration. Nearly
flat spectra are observed below 210 Hz for levels above 130 dB, giving a
dynamic range of about 32 dB. The flat spectrum shape is a significant
improvement over the performance of the old configuration as shown in Figure
2. Standing waves are evident at frequencies of 210, 340 and 480 Hz. For this
reason, the full section operation is limited to less than 210 Hz or to the 220-
330 and 370-480 Hz frequency bands. The effect of standing waves are
explored in further depth in the next section. The spectra in Figure 12 indicate
a nearly uniform distribution in the x-direction throughout the test section. It is
interesting to note that Figure 13 shows no sign of standing waves upstream of
the test section, confirming that the cause is associated with the test section.
Lastly, a near perfect coherence between upstream and downstream, and upper
and lower test section locations is shown in Figure 14 for frequencies between
40 and 210 Hz. Again, this is a significant improvement over the performance
of the old configuration (Figure 3).
Figure 10: Normalized input spectrum Figure 11: Min to max SPL at location
(2-modulator full). 1 (2-modulator full).
925
Figure 12: SPL in test section at max Figure 15: Normalized input spectrum
level (2-modulator full). (4-modulator full).
Figure 13: SPL along length of TAFA Figure 16: Min to max SPL at location
(2-modulator full). 1 (4-modulator full).
Figure 14: Test section coherence (2- Figure 17: SPL in test section at max
modulator full). level (4-modulator full).
The four-modulator full configuration exhibits similar behavior as the two-
modulator full configuration as seen in Figures 16-19. The lowest level at
which a uniform spectrum is achieved is 137 dB, giving a dynamic range of
roughly 30 dB in this configuration. Lastly, the eight-modulator full
926
configuration results, shown in Figures 21-24, indicate a noise floor of about
142 dB and dynamic range of 22 dB.
Frequency, Hz Frequency, Hz
Figure 18: SPL along length of TAFA Figure 21: Min to max SPL at location
(4-modulator full). 1 (8-modulator full).
Frequency, Hz Frequency, Hz
Figure 19: Test section coherence (4- Figure 22: SPL in test section at max
modulator full). level (8-moduiator full).
Frequency, Hz Frequency, Hz
Figure 20: Normalized input spectrum Figure 23: SPL along length of TAFA
(8-modulator full). (8-modulator full).
927
Frequency, Hz Frequency, Hz
Figure 24: Test section coherence (8- Figure 27: SPL in test section at max
modulator full). level (2-modulator reduced).
Figure 25: Normalized input spectrum Figure 28: SPL along length of TAFA
(2-modulator reduced). (2-modulator reduced).
Figure 26: Min to max SPL at location Figure 29: Test section coherence (2-
1 (2-modulator reduced). modulator reduced).
The reduced test section configurations are used to increase the frequency
range and maximum sound pressure level in the test section. Results for the
two-modulator reduced configuration, shown in Figures 26-29, indicate a
nearly flat spectrum between 40 and 480 Hz, a noise floor of 129 dB and a
dynamic range of about 28 dB. Coherence in the test section is nearly unity
928
over this frequency range. This represents a significant improvement over the
old facility configuration. Results of similar quality indicate a d5mamic range
of roughly 26 and 29 dB for the four- (Figures 31-34) and eight-modulator
(Figures 36-39) configurations, respectively. Note that the coherence for these
configurations is slightly reduced at the high frequencies, but is still very good
out to 480 Hz.
Figure 30: Normalized input spectrum Figure 33: SPL along length of TAFA
(4-modulator reduced). (4-modulator reduced).
160 r
80
OASPLs: 134.1, 135.5, 142.1, 147.6,
153.9, 160.1, 165.9, 167.9
1.0
! 1
1 1
Loc 1, Loc 2
Loc 5, Loc 25
400
100
200 300
Frequency, Hz
400
500
Figure 3 1 : Min to max SPL at location Figure 34: Test section coherence (4-
1 (4-modulator reduced). modulator reduced).
Figure 32: SPL in test section at max Figure 35: Normalized input spectrum
level (4-modulator reduced). (8-modulator reduced).
929
80
OASPLs; 134.1, 135.7, 141.9, 148.3,
154.1. 160.0, 165.9. 170.5
80
Loc30
Loci
100 200 300 400 500 100 200 300 400 500
Frequency. Hz Frequency, Hz
Figure 36: Min to max SPL at location Figure 38: SPL along length of TAFA
i (8-modulator reduced). (8-modulator reduced).
Figure 37: SPL in test section at max Figure 39: Test section coherence (8-
level (8-modulator reduced). modulator reduced).
Table 2 presents a summary of the maximum average OASPL for each facility
configuration. In each case, the number of active modulators were run at
maximum power as an independent group (independently for the single
modulator case) and the results averaged. For example, results for one active
modulator were obtained by running each modulator individually and
averaging the resulting pressures.
Table 2: Summary of maximum average overall sound pressure levels (dB).
Number of Active Modulators
1
2
4
8
2-Modulator Red.
1
1 j
2-Modulator Full
i
4-Modulator Red.
159.1
—
MSM
4-Modulator Full
155.6
161.2
8-Modulator Red.
158.4
—
mmm
171.7*
8-Modulator Full
153.0
158.4
164.5
170.0
^Pressure scaled by ^7? from 7-modu
ator run
DISCUSSION
In this section, limiting behaviors of the full and reduced test section
configurations are explored and the effect of test section inserts, modulator
coupling and wave tube performance are discussed.
Limiting Behaviors
The auto-spectra from the full test section configurations exhibit sharp
reductions in level at approximately 210, 340, and 480 Hz. This behavior
corresponds to measurements near nodes of vertical (height) standing waves in
the test section portion of the wave tube. Table 3 summarizes theoretical,
resonant frequencies and corresponding modal indices of the test section duct
resonances within the excitation bandwidth. The modal indices m and n
correspond to half wavelengths in the vertical and transverse (width) directions
of the cross section, respectively. There are several resonances that may be
excited below 500 Hz, but only three of these appear to be significant at the
test section transducer locations (about the horizontal centerline). Because of
the presence of air flow in the facility and lack of measurements in the cross
section, it is difficult to correlate the experimental and theoretical modes.
Measurements of the acoustic pressure at several locations in a cross-section of
the duct will be necessary to fully characterize the resonant behavior. It is
sufficient to say that the usable frequency range in the full test section
configurations is approximately 40-210 Hz near the horizontal centerline.
Acoustic pressure auto-spectra from the reduced test section configurations are
essentially flat to almost 500 Hz. This is due to the fact that only two
resonances are within the excitation bandwidth for this configuration, see
Table 3. A sharp reduction is noted in the vicinity of 480 Hz. Although the
(m=l, n=0) resonance does not appear to be significant, close inspection of the
data (not shown) indicates its presence. Therefore, the usable frequency range
for the reduced test section configurations is approximately 40-500 Hz.
Table 3: Theoretical resonant frequencies of test section duct modes in Hz.
Performance of Test Section Configurations
For constant input acoustic power, the change from full to reduced test section
configurations should theoretically result in a 4,7 dB increase in OASPL.
However, Table 2 shows that increases of only 2.1 (e.g. 164.3-162.2), 0.9, and
1.7 dB were realized for the two-, four- and eight-modulator configurations.
The system efficiency (actual/expected mean-square pressure) of the two-,
four- and eight-modulator reduced configurations is 38, 40 and 44 percent,
respectively, compared with 51, 63 and 62 percent for the two-, four- and
eight-modulator full configurations. The expected pressure is calculated based
upon a input-scaled value of the rated acoustic power of the WAS 3000
modulator assuming incoherent sources (3 dB per doubling). In general, the
full section efficiency is greater than the corresponding reduced section
efficiency. While the reason for this phenomena is not known, it is
conjectured that the lack of expansion in the reduced configurations limits the
development of plane waves. Therefore, phase and amplitude mismatches
between acoustic sources may be accentuated.
Modulator Coupling Performance
A simplified waveguide analysis for coherent, phase-matched sources predicts
increases in OASPL as shown in Table 4. Measured performance gains were
less than predicted because of the assumptions of the waveguide analysis
(inactive source area treated as hard wall), and possible reductions due to
phase differences between modulators and non-parallel wave fironts at the exit
of the modulator cart, see Figures 4-9. The latter effect is due to different
angles of inclination of the sources relative to the axis of the wave tube. The
greater gains achieved in the full test section configurations support the above
contention that they are more efficient than the reduced configurations in
combining the acoustic sources.
Table 4: Change in SPL (dB) from 1 to max. number of active modulators.
Configuration
A SPL fi:om 1 Active Mod. (Meas/Pred)
2-Modulator Red. (2 active mods.)
3.9/6.53
2-Modulator Full (2 active mods.)
5.5/6.53
4-Modulator Red. (4 active mods.)
8.8 / 13.98
4-Modulator Full (4 active mods.)
11.4/13.98
8-Moduiator Red. (8 active mods.)
13.3/22.10
8-Moduiator Full (8 active mods.)
17.0/22.10
Wave Tube Performance
A change in configuration from the two- to the four-modulator configurations,
and from the four- to the eight-modulator configurations, will result in an
incremental increase of 3 dB in OASPL if the individual sources are phase-
matched. This is due to a pure doubling of the power without any change in
the radiation impedance of the individual sources. For the reduced
configurations, a 3.6 and 3.8 dB increase are observed, respectively. A 4.8 and
932
3.0 dB increase are observed for the full configurations, respectively. Note
that a greater than 3 dB increase is possible when the higher modulator
configuration (for example, the four-modulator reduced configuration) is less
susceptible than the lower modulator configuration (the two-modulator
reduced configuration) to phase mismatches between modulators. This seems
plausible because any such mismatches are averaged over a larger number of
sources.
SUMMARY
Modifications to the NASA Langley TAFA facility resulted in significant
improvements in the quality and magnitude of the acoustic excitation over the
previous facility. The maximum OASPL was increased by over 6 dB (vs the
previous 165 dB) with a nearly flat spectrum between 40-210 and 40-480 Hz
for the full and reduced test section configurations, respectively. In addition,
the coherence over the test section was excellent. These improvements,
however, did not meet the objective for a maximum OASPL of 178 dB.
There are several reasons why the maximum OASPL did not meet the
objectives, including a lack of expansion in the reduced configurations and
phase differences between modulators. A detailed computational analysis
would be desirable to indicate the source of the inefficiencies and to help
identify possible means of increasing the overall system performance.
ACKNOWLEDGEMENTS
The authors wish to thank Mr. H. Stanley Hogge and Mr. George A. Parker for
their support in configuring and running the facility. We wish to also thank
Mr. James D. Johnston, Jr. of NASA Johnson Space Center for loan of four
Wyle air modulators.
REFERENCES
1. Maestrello, L., Radiation from a Panel Response to a Supersonic
Turbulent Boundary Layer, Journal of Sound and Vibration, 1969,
10(2), pp. 261-295.
2. Pozefsky, P., Blevins, R.D., and Langanelli, A.L., Thermal-Vibro-
Acoustic Loads and Fatigue of Hypersonic Flight Vehicle Structure,
AFWAL-TR-89-3014,
3. Williams, L.J., HSCT Research Gathers Speed, Aerospace America,
April 1995, pp. 32-37.
4. Clevenson, S.A. and Daniels, E.F., Capabilities of the Thermal
Acoustic Fatigue Apparatus, NASA TM 104106, February 1992.
933
SONIC FATIGUE CHARACTERISTICS OF HIGH TEMPERATURE MATERIALS AND
STRUaURES FOR HYPERSONIC FLIGHT VEHICLE APPLICATIONS-
Dr. I. Holehouse, Staff Specialist,
Rohr Inc., Chula Vista, California
1. INTRODUCTION SUWiARY
A combined analytical and experimental program was conducted to investigate
thermal -acoustic loads, structural response, and fatigue characteristics of
skin panels for a generic hypersonic flight vehicle. Aerothermal and
aeroacoustic loads were analytically quantified by extrapolating existing
data to high Mach number vehicle ascent trajectories. Finite-element
thermal and sonic fatigue analyses were performed on critically affected
skin panels. High temperature random fatigue shaker tests were performed
on candidate material coupons and skin-stiffener joint subelements to
determine their random-fatigue strength at high temperatures. These were
followed by high temperature sonic fatigue tests of stiffened-skin panels
in a progressive wave tube. The primary materials investigated were
carbon-carbon and silicon-carbide refractory composites, titanium metal
matrix composites and advanced titanium alloys. This paper reports on the
experimental work and compares measured frequencies and acoustically
induced response levels with analytically predicted values.
The coupon shaker test data were used to generate material random fatigue
"S-N" curves at temperatures up to 980°C. The joint subelements provided
data to determine the effects on fatigue life of skin-stiffener joining
methods. The PWT sonic fatigue panel tests generated response and fatigue
life data on representative built-up skin panel design configurations at
temperatures up to 925“C and sound pressure levels up to 165 dB. These^
data are used in determining the response strains and frequencies of skin
panel designs when subjected to combined thermal -acoustic loading and to
identify modes of failure and weaknesses in design details that affect _
sonic fatigue life. Sonic fatigue analyses of selected test panel design
configurations using finite-element techniques were also performed and
related to the experimental results. Acoustically induced random stresses
were analytically determined on a mode-by-mode basis using finite element
generated mode shapes and an analytical procedure that extends Miles
approach to include multi-modal effects and the spatial characteristics of
both the structural modes and the impinging sound field.
The paper also describes the instrumentation development work performed in
order to obtain reliable strain measurements at temperatures in excess of
conventional strain gauge capabilities. This work focused primarily on the
use of recently developed high temperature (350"C to 1000“C) strain gauges,
laser Doppler vibrometers, high temperature capacitance displacement
probes, and the determination of strain-displacement relationships to
facilitate the use of double integrated accelerometer data to derive strain
levels.
935
This work was funded by the USAF Flight Dynamics Laboratory (Kenneth R.
Wentz, Project Engineer). The complete program report is contained in
References 1 and 2.
2. HIGH TEMPERATURE STRAIN MEASUREMENTS
Conventional adhesively bonded strain gauge installations are temperature
limited to approximately 350°C. In order to achieve strain measurements at
higher temperatures, up to QSO^C, ceramic layers and coatings were used to
both attach strain gauges and to thermally protect them. However, such
strain gauge installations are very sensitive to process parameters which
often need varying depending upon the test specimen material. Coated
carbon-carbon is a particularly difficult material to adhere to due to its
material characteristics and relatively rough surface texture. Carbon-
carbon also has a near zero coefficient of thermal expansion which presents
attachment and fixturing problems in a high temperature environment.
When high test temperatures either preclude or make problematic the use of
strain gauges, an alternative technique for obtaining strain levels is to
measure displacements and then determine strain levels using strain-
displacement ratios. Strain is directly proportional to displacement for a
given deflected shape, or mode shape, regardless of changes in the elastic
modulus of the specimen material as it is heated. Consequently, if the
deflected shape does not change significantly with temperature, high
temperature test strain levels can be determined from room temperature
strain and displacement measurements in combination with displacement
measurements made at the test temperature.
This measurement technique facilitates the use of non- contacting
transducers which can be located away from the heated area, such as
capacitance displacement probes or Laser Doppler Vibrometers (LDV). LDVs
actually measure surface velocity but their signal outputs can be readily
integrated and displayed as displacement. Accelerometers can also be used
to measure displacement by double integrating their signal output.
However, since accelerometers require surface contact they have to either
withstand, or be protected from, the thermal environment. When this is not
readily achievable, it is sometimes possible to install an accelerometer at
a location on the test specimen or fixturing where the temperature is
within its operating range, providing the displacement response at the
point of measurement is fully coherent with the strain response at the
required location.
The displacement range limitations of the LDV and capacitance probes
available to the program resulted in having to use double-integrated
accelerometer outputs to measure displacements at room temperature and at
the test temperature. Conventional strain gauges were used to measure
strains at room temperature. In order to confirm that the strain-
displacement ratios were unaffected by temperature, limited high
temperature strains were measured at temperatures up to 980* C. Once the
strain-displacement ratio for a given specimen type was determined, air¬
cooled accelerometers were used to determine high temperature test strain
levels. The level of measurement accuracy of this technique was estimated
to be within 10 percent.
936
The most successful strain measurements made at 980“ C utilized a ceramic
flame spray installation of an HFN type free filament gauge. This gauge
installation included the use of silicon-carbide (SiC) cement as a base
coat for the gage, applied over a 1-inch square area of a lightly sanded
carbon-carbon surface substrate. Lead wire attachments to the gauge were
made with standard Nichrome ribbon wire anchored to the specimen with SiC
cement. With this gauge installation, it was possible to make dynamic
strain measurements for short periods of time at 980“C.
3. RANDOM FATIGUE SHAKER TESTS
The instrumented test specimens were mounted in a duckbill fixture and the
specimen/fixture assembly then enclosed in a furnace. An opening in the
furnace allows the specimen tip to protrude out in order to accorrmodate the
air-cooled tip accelerometer. Figure 1 shows strain gauge locations and
fixturing for material coupon and joint subelement specimens.
The test procedure comprised a room temperature sine-sweep in order to
identify the fundamental mode and its natural frequency, one-third octave
random loading at room temperature centered around the fundamental natural
frequency and one-third octave random endurance testing at the required
test temperature and load level.
Twelve inhibited carbon-carbon material coupons generated usable S-N data,
eleven at 980“C and one at 650“C. S-N data points were also generated at
980°C for two integral joint and two mechanically fastened joint
subelements. Fixturing problems and specimen availability limited the
number of S-N data points generated. Figure 2 shows the random fatigue S-N
data points with joint subelement data points superimposed. The random
fatigue endurance level for the material coupons, extrapolated from 10 to
10® cycles, is approximately 320 microstrain rms. The integral joint
subelements did not fail at the strain gauge locations; consequently, the
actual maximum strain levels were higher than those shown on Figure 2.
Taking this into account, it appears that the integral joints have a
fatigue endurance level of greater than one-half of that for the material
coupons. The mechanically fastened joint subelements exhibited fatigue
strength comparable to that of the material coupons. These results
indicate that carbon-carbon joints and attachments methods are not
critically limiting factors in the structural applications of inhibited
carbon-carbon. Figure 3 shows a representative example of the strain
amplitude and peak strain amplitude probability density functions at room
temperature for a material coupon specimen. The "peak" function can be
seen to approximate a Rayleigh distribution, as it should for a Gaussian
random process.
Random fatigue S-N data were also generated for enhanced silicon-carbide
composites (SiC/SiC) including thermally exposed specimens (160 hours at
980“ C), titanium metal matrix composites (TMC) utilizing Ti 15-3 and Beta
21S titanium matrix materials, titanium aluminide (super alpha two),
titanium 6-2-4-2, titanium 6-2-4-2-$i (including thermally exposed
specimens) and Ti-1100. The fatigue endurance levels are shown in Table 1.
Also shown in Table 1 are S-N data points for uninhibited carbon-carbon
generated on a previous program (Reference 3).
937
TABLE 1. SUMMARY OF RANDOM FATIGUE ENDURANCE LEVELS.
material
TEMPERATURE
ENDURANCE LEVEL CORRESPONDING TO 10®
CYCLES: OVERALL RMS STRAIN (MICROSTRAIN)
material COUPONS
SUBELEMENTS
INHIBITED CARBON-CARBON
1800“F (980"C)
320
integral JOINTS > 160
BOLTED JOINTS 320
5 PLY
1800T (980"C)
100
-
UNINHlbl 1 tU
CARBON-CARBON
1ft PI Y
ISOO'-F igSOT)
150
_
*5 PLY
1000‘'F I540"C)
100
-
18 PLY
lOOO'F (540-C)
450
-
5 PLY
ROOM TEMPERATURE
550
-
18 PLY
ROOM TEMPERATURE
450
-
ENHANCED
SiC/SiC
NON-EXPOSED
1800"F r980“C)
450
-
THERMALLY EXPOSED
iaOO“F f980‘'C)
300
-
1000“F (540*0
520
DIFFUSION-BONDED HAT-
STIFFENED = 520
n IML
ROOM TEMPERATURE
2250
-
beta 21S TMC
200
510
-
TITANIUM ALUMINIDE
/ciinCD ftl DUIA TUn^
ROOM TEMPERATURE
410
{ rvui 1 iri • *
Ti-6242-Si
735
Hilllll
LIO BONDED HONEYCOMB
BEAM = 388
■■
WELDED JOINT = 400
T-; nnn
-
Ti 6-2-4-2
ROOM TEMPERATURE
675
-
866MISC/039-T1.IH
12-02-96
938
Figure 4 shows random fatigue S-N curves for the materials tested
superimposed on one graph for comparison purposes. The Ti 15-3 TMC data
are not shown since this was a concept demonstrator material utilizing a
Ti 15-3 matrix material for producibility reasons. Ti 15-3 does not have
the temperature capability for hypersonic vehicle applications. Titanium
aluminide data are not shown due to its brittle material characteristics
making it unsuitable for sonic fatigue design critical structures. Ti-1100
S-N data were very similar to the non-exposed Ti 6-2-4-2-Si and are not
shown. Ti 6-2-4-2 coupons were only tested at room temperature before
beingVeplaced by Ti 6-2-4-2-Si, which has higher structural temperature
capabilities.
The fatigue curves in Figure 4 show inhibited carbon-carbon to have higher
fatigue strength at 980°C than does its uninhibited counterpart. Inhibited
carbon-carbon also has greater resistance to oxidation at high
temperatures.
Although unexposed enhanced SiC/SiC had greater random fatigue strength at
980“C than did inhibited carbon-carbon, the two materials exhibited similar
strength at temperature after allowing for thermal exposure. However,
SiC/SiC has a maximum temperature capability of 1100 to 1200°C compared to
1700 to 1900“C for carbon-carbon.
The Beta 21S TMC material demonstrated resonable fatigue strength at 815°C
and the Ti 6-2-4-2-Si specimens exhibited high fatigue strength at 620°C to
650“C.
4. SONIC FATIGUE PANEL TESTS
These tests were performed in Rohr's high temperature progressive-wave tube
(PWT) test facility. The facility is capable of generating overal 1^ sound
pressure levels of 165 to 168 dB at temperatures up to 925 “C to 980“ C,
depending upon the test panel configuration and material.
Three rib-stiffened carbon-carbon panels and a monolithic hat-stiffened
Beta 21S TMC panel were subjected to sonic fatigue testing. Response
strains were measured on the four panels over a range of incrementally
increasing sound pressure levels (140 to 165 dB) at room temperature. One
carbon-carbon panel was subjected to sonic fatigue testing at room
temperature and the other two tested at 925“ C. The TMC panel was endurance
tested at 815“C. Figures 5 and 6 show a carbon-carbon panel and its
fixturing installed in the PWT. The panels were attached to the fixture
via flexures in order to allow for differences in the thermal expansion of
the panel and fixture materials. Structural details of the panels and
instrumentation locations are given in References 1 and 2.
The three carbon-carbon panel configurations encompassed two skin
thicknesses and two stiffener spacings as follows:
Panel 1: 3 skin bays, 6 in. by 20 in. by 0.11 in. thick
Panel 2: 2 skin bays, 9 in. by 20 in. by 0.11 in. thick
Panel 3: 3 skin bays, 6 in. by 20 in. by 0.17 in. thick
939
Table 2 summarizes the measured room temperature frequencies and strain
response levels:
TABLE 2. ROOM TEMPERATURE RESPONSE OF TEST PANELS-
TEST
PANEL
FREQUENCY OF
IN-PHASE
MODE (Hz)
OASPL
(dB)
OVERALL RMS STRAINS
rMICROSTRAIN)
EDGE OF
SKIN BAY
CENTER OF
SKIN BAY
CARBON-CARBON NO. 1
267
165
305
149
155 & 171
145
126
39
191
59
165
558*
173*
CARBON-CARBON NO. 3
423
165
69
127
BETA 21$ TMC
241
165
HIGHEST STRAIN = 287
AT PANEL CENTER ON
STIFFENER CAP
* EXTRAPOLATED ON THE BASIS OF TUE STRAIN RESPONSE WITH SPL FOR
PANELS 1 AND 3.
Panel 1 was subjected to 165 dB at room temperature for 10 hours at which
point cracks developed at the ends of the stiffeners. The frequency
dropped slightly during the ten hour test resulting in the number of cycles
to failure being approximately 9 million.
Panel 2 was endurance tested at 925“C at 150, 155 and 160 dB for 3-1/2
hours at each level, followed by one hour at 165 dB. At this point, cracks
were observed at the ends of the stiffeners, similar to the cracks in
Panel 1.
Panel 3 was endurance tested at 925“C and 165 dB for 10 hours without any
damage to the panel.
The TMC panel was endurance tested at 815° C and 165 dB for 3-1/2 hours at
which time cracks were observed in two stiffener caps at the panel center.
The high test temperatures for Panels 2 and 3 and the TMC panel precluded
attaching an accelerometer directly to the panel surface, even with air
cooling. This prevented the direct measurement of panel displacements at
925° C. In order to attempt to estimate the high temperature endurance test
strain levels, a temperature survey was performed on the panel fixturing
with Panel 3 installed in order to determine an acceptable location for an
accelerometer. An accelerometer at the selected fixture location tracked
linearly with the highest reading strain gauges during a room temperature
response survey. The coherence between the fixture accelerometer and the
panel strain gauges was 0.9 in the frequency range of panel response.
940
Having established a coherent strain displacement relationship at room
temperature, the temperature was increased progressively with increasing
acoustic loading, generating accelerometer and microphone data at 480 C and
140 dB, 650°C and 155 dB, 860°C and 155 dB, and 925“C at 165 dB. It was
clear from the data at the higher temperatures and load levels that the
full spectrum overall rms displacement levels obtained by double integrat¬
ing the accelerometer output signals could not be used to determine high
temperature strain levels due to high amplitude, low frequency displace¬
ments (displacement being inversely proportional to frequency squared for a
given "g" level) that were well below the panel response frequency range
and therefore would not be proportional to panel strain levels. It is
important to remember here that since the accelerometer is mounted on the
panel fixture, it is measuring fixture response, some of which is not
related to panel response.
After reviewing the various frequency spectra, it was decided to re-analyze
the data to generate overall rms levels over selected frequency bandwidths
that would encompass a high percentage of the full -spectrum overall rms
strains and eliminate the low frequency displacements. If a consistent
strain-displacement relationship could be established at room temperature
within a frequency bandwidth such that the strains could be related to the
full-spectrum overall rms strains, and if the same bandwidth could be used to
generate displacements at temperatures that were sufficiently consistent to
relate to strain response, then it would be possible to at least make a
reasonable estimate of the test temperature strain level. It was determined
that band-passed response data in the 300 to 600 Hz frequency range gave
consistent strain-displacement ratios at room temperature. Double-integrated
band-passed accelerometer outputs (displacements) were consistent with
increasing sound pressure levels at incrementally increasing test temperatures
up to the 925°C/165 dB endurance test conditions. Table 3 summarizes the high
temperature test panel results.
TABLE 3. HIGH TEMPERATURE TEST PANEL RESULTS.
TEST PANEL
TEST
TEMPERATURE
OVERALL
SOUND
PRESSURE
LEVEL (dB)
HIGHEST
ESTIMATED
OVERALL RMS
STRAIN
(MICROSTRAIN)
EXPOSURE TIME,
ESTIMATED FATIGUE
CYCLES AND COMMENTS
(°F)
rc)
BETA 21S TMC
PANEL
1500
815
165
NOT ESTIMATED
3 1/2 HRS, 3x10“ CYCLES,
STIFFENERS CRACKED AT
MID-SPAN
CARBON-CARBON
PANEL NO. 2
150
155
TTTORSTTTMo^TYOlsr
NO FAILURE
155
219 ^
3 1/2 HRS, 2.3x10° CYCLtS,
NO FAILURE
160
316
3 1/2 HRS. 2.3x10“ CYCLES,
NO FAILURE
165
453
1 HR, 6.4xl0‘> cycles!
CRACKS AT STIFFENER ENDS
925
165
103
10 HRS, 1.7x10' CYCLES,
NO FAILURE
941
It should be noted that carbon-carbon panels 1 and 2 exhibited cracks at
the stiffener ends, whereas the maximum measured strains were at the edges
of skin bays. Consequently, the actual strain levels at the crack
locations were either higher than the measured levels or there were
significant stress concentrations at the stiffener terminations.
5. COMPARISON OF ANALYTICAL AND TEST RESULTS FOR CARBON-CARBON PANELS
MSC NASTRAN was used to perform finite element analyses on the three
carbon-carbon panels that were subjected to the sonic fatigue testing
described in Section 4. The oxidation resistant coating was modeled as a
non-structural mass, which is compatible with the panel test results. ^
Natural frequencies, mode shapes and acoustically induced random strain
levels were analytically determined for room-temperature conditions and
compared to the room-temperature panel test results.
Acoustically induced random stresses were analytically determined on a
mode-by-mode basis using the finite element generated mode shapes and a
Rohr computer code based on an analytical procedure presented in
Reference 4. This procedure extends Miles* approach (Reference 5) to
include multi-modal effects and the spatial characteristics of both the
structural modes and the impinging sound field.
Table 4 shows the calculated and measured frequencies, overall rms strain
levels and the strain spectrum levels for the in-phase stiffener bending
mode for the carbon-carbon panels at room temperature.
TABLE 4. CALCULATED AND MEASURED RESPONSE FREQUENCIES AND STRAIN
LEVELS FOR CARBON-CARBON PANELS AT ROOM TEMPERATURE.
STRAIN LEVELS AT EDGE OF SKIN BAY
NATUF
FREQUE
OF IN-F
MODE
(Hzl
\fKL
:ncy
>HASE
OVERALL RMS STRAIN
(MICROSTRAIN)
STRAIN
SPECTRUM LEVEL
IN-PHASE MODE
(MICROSTRAIN/Hz)
FE ANALYSIS
MEASURED
FE ANALYSIS
MEASURED
FE ANALYSIS
MEASURED
PANEL 1
a65 dBl
305
267
510
305
84
60
PANEL 2
(145 dBl
190
155 &
171
133
126
40
41
PANEL 3
(165 dB)
460
423
77
69
16
16
942
The above results show good agreement between the finite element generated
values and those measured. The level of agreement is particularly good for
the strain spectrum levels, which are typically more difficult to
accurately predict. Figure 7 shows the finite-element frequency solution
for Panel 3. The in-phase mode shape can be seen to have an overall modal
characteristic due to the relatively low bending stiffness of the
stiffeners for the skin thickness used. Figure 8 shows the measured and
finite-element generated strain frequency spectra for Panel 3.
Details of the finite-element analyses and models are contained in
References 1 and 2.
6. CONCLUSIONS AND RECOMMENDATIONS
1. The high temperature testing techniques and strain measuring
procedures successfully generated usable random fatigue S-N
curves and panel response data. The use of strain-displacement
ratios were shown to be an effective alternative to high
temperature strain gauge measurements.
2. In general, the materials and structural concepts tested
demonstrated their suitability for hypersonic flight vehicle skin
panel applications. The major exception was Titanium-Aluminide
Super Alpha Two which was determined to be too brittle.
3. Inhibited carbon-carbon exhibited significantly higher random
fatigue strength at 980°C than did the uninhibited carbon-carbon
— two to three times the random fatigue endurance strain level.
4. Thermally exposed enhanced SiC/SiC had comparable fatigue
strength to that of inhibited carbon-carbon at 980 °C.
5. The TMC specimens usefully demonstrated the fatigue strength of
the TMC concept and the need to develop the concept to
incorporate higher temperature capability titanium matrix
materials.
6. Titanium 6-2-4-2-Si exhibited high fatigue strength in the 590°C
to 650 “C temperature range and also demonstrated the need for TMC
materials to utilize higher temperature matrix materials in order
to be cost effective against the newer titanium alloys.
7. The level of agreement between the finite element analysis
results for the carbon-carbon panels and the progressive-wave
tube test data demonstrated the effectiveness of the analytical
procedure used. The analysis of structures utilizing materials
such as carbon-carbon clearly presents no special difficulties
providing the material properties can be well defined.
943
8 It is recommended that further tests be conducted similar to
those performed in this program but with greater emphasis on
testing panels having dimensional variations in order to develop
design criteria and life prediction techniques. Such testing
should be performed on those structural materials and design
concepts that emerge as the major candidates for flight vehicle
applications as materials development and manufacturing
techniques progress.
REFERENCES
1 R D. Blevins and I. Holehouse, "Thermo-Vibro Acoustic Loads and
rkigue of Hypersonic Flight Vehicle Structure," Rohr, Inc.
Engineering Report RHR 96-008, February 1996.
2. United States Air Force Systems Command, Flight Dynamics
Laboratory Final Technical Report, Contract No. F33615-87-C-33^^/,
to be published.
3. R. D. Blevins, "Fatigue Testing of Carbon-Carbon Acoustic Shaker
Table Test Coupons," Rohr, Inc. Engineering Report RHR 91-087,
September 1991.
4. R. D. Blevins, "An Approximate Method for Sonic Fatigue Analysis
of Plates and Shells," Journal of Sound and Vibration, Vol. 129,
51-71, 1989.
5. J. W. Miles, "On Structural Fatigue Under Random Loading,"
Journal of Aeronautical Sciences, Vol. 21, November 1954.
944
A. Test Configuration for
Material Coupons
Fixture
Specimen
■ O
B. Test Configuration for
Carbon-Carbon Integral
Stiffener Specimens
Accelerometer
t
C. Test Configuration For
■ Carbon-Carbon Mechanically
Fastened Stiffener and All
Titanium Diffusion Bonded
Joint Specimens
FIGURE 1 Typical Strain Gauge Locations and Test Configurations
for Material Coupon and Joint Subelement Shaker Test
Specimens
945
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950
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SCALING CONCEPTS IN RANDOM ACOUSTIC FATIGUE
BY
Marty Ferman* and Howard Wolfe**
ABSTRACT
Concepts are given for scaling acoustic fatigue predictions for application to
extreme environmental levels based on testing “ scaled” structures at
existing, lesser environmental levels. This approach is based on scaling a test
structure to fit within the capabilities of an existing test facility to attain
fatigue results, and then using analytical extrapolation methods for predicting
the full scale case to achieve accurate design results. The basic idea is to
utilize an existing acoustic fatigue facility to test a structure which has been
designed (scaled) to fatigue within that facility’s limits, employing the
appropriate structural properties ( such as thinning the skins, etc.). Then, the
fatigue life of the actual structure is determined by analytically scaling the test
results to apply to the full scale case ( thicker) at higher noise levels for
example. Examples are given to illustrate the approach with limits suggested,
and with the recognition that more work is needed to broaden the idea.
BACKGROUND
While it is important to continually expand the capability of acoustic test
facilities , it is perhaps equally important to be able to work with existing
facilities at any time. That is, facility expansions, enhancements , and
modernization’s should always be sought from time to time, so long as
practical and affordable from cost effective considerations. Limits should be
pushed to accommodate larger sizes of test specimens with higher noise levels
with wider ranges of frequencies, with wider ranges of temperatures, and with
better capabilities for applying pressures along with any one of several types
of preloads. These are costly considerations and require considerable time to
accomplish. Facility rental can be used in some cases to bolster one’s testing
facilities, however if the application suggests a situation beyond any available
facility for the required design proof, then an alternate is needed. Thus the
scaling concept suggested here is a viable and useable possibility.
The Author’s basis for the approach stems from their extensive, collective,
experience in Structural Dynamics, especially work in Acoustic Fatigue,
Fluid-Structure Interaction, Buffet , and Aeroelasticity/Flutter , and
particularly from experience with flutter model testing, in which it is quite
common to ratio test results from a model size to full scale for valid
Assoc. Prof, Aerospace and Mech. Engr. Dept, Parks College, St. Louis
Univ., Cahokia, IL, 62206, USA
** Aerospace Engineer, Wright Laboratory, Wright Patterson AFB, OH,
45433, USA
953
predictions . Flutter is a well recognized area where model data is commonly
used in nondimensional form to establish design margins of safety, as typified
in Ref (1-2). Flutter can be nondimensionalized quite broadly as pointed out
in many works, and is clearly done for a wide range of general cases using the
“so-called” Simplified Flutter Concept, Ref (3) . The degree of the use of the
flutter model scaling rules varies considerably today, because some people
are testing as much or more than ever, while others are testing less and relying
more heavily on advanced theories such as Computational Fluid Dynamics,
CFD. However, the basic ideas in flutter model scaling are still POWERFUL!
In fact, this concept has fueled the Author’s desires to develop the
“acoustical scaling” used in the approach presented. Moreover, when starting
to write this paper, the Authors realized that this type of scaling is also
common to many related areas of structural dynamics, and thus chose to
include some examples of those areas to emphasize the main point here!
For example, experience in fluid-structure interaction and fatigue of fuel tank
skins, a related work area, serves as another example of scaling structures to
demonstrate accurate predictions with widely varied environmental levels,
and a multitude of configurations. Scaling and nondimensional results were
used extensively in Ref (4 -13), and are cited here because of the immense
data base accumulated. The work at that time did not necessarily define
scaling as used here , but hindsight now suggests that there is a clear relation.
It is becoming well recognized that Buffet is easily scaled , and many
engineers and investigators are now employing scaling of pressures from
model to full size applications, and are also using scaled model response to
predict full scale cases . Some of the earliest and some of the more modern
results clearly show this aspect. For example. Ref (14 -16) are typical, quite
convincing, and pace setting regarding scaled data. Buffet models which are
much more frail that the full-scale cases are used to develop data for full scale
applications, and besides giving full sized results, provide a guide to safe
flight testing as has been done more extensively with flutter testing.
Obviously, acoustical response and fatigue phenomenon are also
nondimensionalizable and scaleable. Ref (17-18), for example. This point is
being taken further here; that is , scaling will be used to take better advantage
of limited facility testing capability to predict more severe situations, as is
used in the case of flutter model testing where a larger specimen is predicted
from tests of a smaller structure using similarity rules. Here in the acoustic
application, a thinner , or otherwise more responsive specimen, is tested and
then analytical means are used to make the prediction for the nominal case.
APPROACH
The method is shown here is basically an extension of the flutter model
scaling idea, as applied to acoustical fatigue testing with a particular emphasis
on random applications. The technique will also work for sine type testing in
954
acoustical fatigue, and perhaps it will be even more accurate there, but most of
today’s applications are with random testing, notably in the aircraft field.
Thus it is in this area where the method should find more application . The
Authors have a combined professional work experience of some 70+ years
and thus have tried to focus this extensive background on an area where gains
can be made to help reduce some costs while making successful designs, by
using lesser testing capability than might be more ideally used. It is believed
that the best testing for random acoustic fatigue, is of course, with (a) the
most highly representative structure, and as large a piece as can be tested,
both practically and economically, (b) the most representative environmental
levels in both spectrum shape and frequency content, (c) test times to
represent true or scaled time, as commonly accepted, (d) temperatures should
be applied both statically and dynamically, and finally (e) preloading from
pressures, vibration, and from boundary loading of adjacent structure.
Frequently, testing is done to accomplish some goal using a portion of these
factors, and the remainder is estimated . Thus the Authors believed that there
is a high potential to extend the flutter model approach to acoustical
applications.
Recall that in the flutter model approach , the full scale flutter speed is
predicted by the rule
((Vf)a)p- [ ((Vi.)m)e/ (( Vf)m)c] X [ ((Vf)a)c] (^)
where Vp is flutter speed, the subscripts M and A refer to model and aircraft
respectively, the subscript C refers to calculated, and the subscript P refers to
predicted. Thus the equation suggests that the full scale predicted flutter speed
is obtained by taking the ratio of experimental to calculated flutter speed for
the model and then multiplying by a calculated speed for the airplane. These
flutter model scaling ideas are covered in any number of References, i.e. Ref
(lo), for example.
The same concept can be utilized in acoustic fatigue, i.e. the strain at fatigue
failure relation, (8,N) can be scaled from model structure tested at one level
and then adjusted for structural sizing and environmental levels. This relation
can be addressed as done for the flutter case:
((e,N)a)p=[(( s,N)m)e / ((e,N)m)c] X [((e.N)a)c] (2)
where s is strain, and N is the number of cycles at failure, where as above in
Eq (1) , the subscripts M and A refer respectively to Model and Full Scale for
parallelism, while the subscripts E, C, and P have the same connotation again,
namely, experimental, calculated, and predicted. Thus the full scale case is
955
predicted from a subscale case by using the ratio of experimental to
theoretical model results as adjusted by a full scale calculation. Flutter model
scaling depends upon matching several nondimensional parameters to allow
the scaling steps to be valid. While these same parameters are, of course, not
necessarily valid for the acoustic relationships, other parameters unique to
this acoustical application must be considered, and will be discussed.
Accurate predictions for the method relies on extensive experience with the
topic of Acoustic Fatigue in general, because concern is usually directed
towards the thinner structure such as; panels, panels and stiffeners, and panels
and frames, bays (a group of panels), or other sub-structure supporting the
panels. These structures are difficult to predict and are quite sensitive to edge
conditions, fastening methods, damping, combination of static and dynamic
loading, and temperature effects. Panel response prediction is difficult , and
the fatigue properties of the basic material in the presence of these complex
loadings is difficult. However, the experienced Acoustic Fatigue Engineer is
aware of the limits, and nonnally accounts for these concerns. Thus the
method here will show that these same concerns can be accounted for with the
scaling approach through careful considerations.
The Authors believe that the method is best explained by reviewing the
standard approach to acoustic fatigue, especially when facility limits are of
major concern. Fig (1) was prepared to illustrate these points of that
approach. Here it is seen that key panels for detail design are selected from a
configuration where the combination of the largest, thinnest, and most
severely loaded panels at the worst temperature extremes and exposure times
are considered. These can be selected by many means ranging from empirical
methods, computational means, and the various Government guides. Ref (17-
18), for example. Then detailed vibration studies are run using Finite
elements , Rayleigh methods. Finite Difference methods, etc. to determine the
modal frequencies and shapes, and frequently linearity is assessed. Then
acoustical strain response of the structure is determined for sine, narrowband,
and broadband random input to assess fatigue life based on environmental
exposure times in an aircraft lifetime of usage. These theoretical studies are
then followed by tests of the worst cases, where vibration tests are conducted
to verify modal frequencies, shapes, and damping, and linearity is checked
again for the principal modes. This is followed by acoustical strain response
tests where the strain growth versus noise levels is checked, again employing
sine, narrowband and broadband random excitation. Note the figure suggests
that data from the vibration tests are fed back to the theoretical arena where
measured data are used to update studies and to correlate with predictions,
especially the effect of damping on response and fatigue, and of course, the
representation of nonlinearity. Also, the measured strain response is again
used to update fatigue predictions. These updates to theory are made before
the fatigue tests are run to insure that nothing is missed. However, in this
956
case, the required sound presssure level SPL in (dB) is assumed to exceed the
test chamber’s capability. Thus , as shown in the sketch in Fig (2) the key
strain response curve, s vs dB, is extrapolated to the required dB level. This
data is merged with the strain-to-failure curve at the right to establish the cycle
count, N, giving the (s, N) point for this case. The extrapolated data provides
some measure of the estimated life, but again is heavily dependent upon the
accuracy of the basic strain response curve, and is especially dependent on
whether high confidence exists at the higher strains. Linear theory is also
shown in this case, indicating it overpredicts the test strain response and hence
shows a shortened fatigue life compared to test data, as is generally the case
in today’s extreme noise levels. This illustration is highly simplified,
because experienced designers readily know that it is difficult to predict even
simple panels accurately at all times, let alone complex and built-up structure
consisting of bays (multi-panel); this will addressed again later in the paper.
The new concept of scaled acoustic fatigue structures is shown on the sketch
of Fig (3) where the standard method is again shown, but smaller in size, to
refocus the thrust of the new idea. The scaling process parallels the standard
approach, and actually complements that approach, so that the two can be run
simultaneously to save time, costs, and manpower. Here the panel selection
process recognizes that the design application requires environments far in
excess of available facility capability. Thus the scaling is invoked in the
beginning of the design cycle. As the nominal panels (bays) are selected and
analyzed for vibration, response and fatigue, scaled structures are defined to
provide better response within the existing chamber ranges so that they can be
fatigued and then the results can be rescaled to the nominal case. In this
manner, appropriate designs can be established to meet safety margins with
more confidence, and will avoid costly redesign and retrofitting at
downstream stages where added costs can occur and where down times are
difficult to tolerate. The concept is further illustrated in the sketch of Fig (4).
Here the strain response curve of the nominal case and that of the scaled
version are combined with strain to failure data ( coupon tests) to show fatigue
results. Note the strain response for the nominal case at the highest dB level
available gives the fatigue value at point A, while the extrapolated data for
this curve gives point B. The scaled model being more responsive gives the
point C, and when rescaled gives the point D which differs slightly from the
extrapolated point B as it most likely will, realistically. More faith should be
placed on data from an actual fatigue point than a point based on the projected
strain response curve. Note, Fig (5) illustrates the winning virtue of the
scaleable design. The figure shows a hypothetical set of test data for the (s ,N)
for a structure for various SPLs for the nominal case, open circles, and for the
fatigue results of the scaled model , closed squares. The scaled model was
assumed to be thinner here for example, and that the scaled data is also
957
rescaled to fit the nominal curve here. The most interesting aspect is shown
by the two clusters of data , denoted as A and B where there are rough circles
about drawn about the clusters. Here the emphasis is that tests of the scaled
model ( and rescaled data ) are used to find the higher strain conditions which
cannot be found from the nominal case. In both cases at the highest strain
levels, the facility is used to its limits, but with enough testing with the thinner
case, adequate data is available to make the prediction more accurate using
Eq (2) for the final correlation as shown here. The statistical scattering of the
scaled data will be an accurate measure for the nominal case, particularly
when compared to estimates based on extrapolation of the strain response for
the nominal case. There are many cautions to be noted with this approach as
there are with all acoustic fatigue methods, and of course, tests. First, the
linearity of the modes, either in unimodal sine excitation, multi-mode sine,
narrow band or broadband random must be carefully handled. The strain
response of individual locations throughout the structure must be carefully
monitored in calculations and tests so that strain response is truly understood
and used to define fatigue life carefully. This is difficult to do in many
applications where widely varying conditions and durations require some type
of Miner Rule combination to provide a true measure of fatigue. Similarly,
strain risers at fasteners, discontinuities, holes, frames, stiffeners, material
changes along with temperature gradients, temperature transients, require
final “tweeks” to predictions, regardless. Nonlinearity , especially in the
multi -mode case, is one of the most formidable foes to conquer for any
application.
APPLICATIONS AND EXAMPLES
The tests of an Aluminum panel of size 10x20 in. and with a thickness of
0.063 thickness , Ref (19), will be used to illustrate the technique. The panel
has approximately fixed-fixed edge conditions, and is quite nonlinear in
terms of strain response. Ref (19). The measured strain response for the panel
is shown in Fig (6) along with an estimated response curve for a thinner panel
( 0.040 in ) based on the test data. For this case it is assumed that data were
needed at 175 dB, while the facility could only achieve 164 dB. The strain
response for the thinner case was estimated using the classical equation for
the amplitude of response, 5 ;
\\fdxdy\pSD,{f)]
(3)
where M is the generalized mass, co is the natural frequency, § is the viscous
damping factor , (j) is mode shape, PSDp is the pressure Power Spectral
Density, and x,y are the positional coordinates along the plate. Since strain ,
s, is proportional to the amplitude.
958
s= (t/2)(3(ti^/ax^)S
(4)
Combining Eq (3 and 4) shows that the strain response curves are
proportional to the thickness factor, as given by:
s 2 ~ ( ti / t2) (5)
However, it must be noted that this case is nonlinear, and thus, this result is
not exactly correct, but simply used for an illustration here. The actual data for
the 0.063 thickness is extrapolated to a required 175 dB, showing a strain of
1000 micro in/in. The estimated curve for the thickness of 0.040 in. shows ,of
course, a greater response at all dB levels as it should, and moreover shows
that only 150 dB are needed to achieve the 1000 micro - strain condition.
Moreover, the thinner panel will exhibit large enough strains at the lower
SPLs to improve the fatigue curve where the thicker panel is insensitive.
Taking the example a step further, the fatigue point of the nominal case is
shov^Ti on a strain to failure plot in Fig (7) , employing beam coupon tests of
Ref (11), which were shown to be excellent correlators with panel fatigue in
the collection of work in Ref (6-13). The fatigue point for the 164 dB
excitation, 800 micro-strain, is shown as a triangle, while the extrapolated
data for 175 dB is shown as the flagged triangle. One test point exists for the
nominal case. Ref (20) , and is shown by a star symbol. Data for the thinner
panel are shown as circles at the various strain to cycle count cases for the
various SPLs corresponding to the beam curve. Notably, these points can be
seen to produce shorter fatigue cases as they should due to increased strains,
but note that they are also at lower frequencies which would give a longer test
time than if they were the nominal thickness. The scaled model is seen to
produce the same point as the extrapolated case in this hypothetical case for
the 1000 microstrain case ( again, at two different dB levels for two
thicknesses). A SPL of 150 dB, rather benign, is seen to be quite effective.
The actual fatigue point at 164 dB for the nominal case required 3 hours and
was predicted to be 2.8 hrs. The estimated fatigue for the extrapolated case of
175 dB was estimated to be 1.7 hrs, while the scaled point from the thinner
panel was estimated to be 2.2 hours which is slightly off, but the Authors
have had to rely on log plots for much of the data and thus lack someaccuracy.
Because of lack of actual data , the scatter from the estimate vs the test of the
nominal case was used to scatter the estimate for the 1000 microstrain case,
flagged dark circle, as if the use of Eq (2) had been employed directly. One
must be careful here, because there can be a vast difference between theory
and test, and this can mislead inexperienced persons applying these methods.
As noted earlier, related work in fluid-structure and buffet , actually
demonstrate this type of scaling. To illustrate, several figures are republished
959
here to make this point rather clear. Fig (8) of Ref (8 ) shows the dynamic
bending strains in the bottom panel of an otherwise rigid fuel tank which is
being excited vertically with moving base input. The vertical axis is strain
while the abscissa is the number of g’s input. Three panel thicknesses and four
depths of fluid (water in this case) were used. Note the sharp nonlinear effect
in the response, rather than linear response growth as force increases.
Interestingly, the data was nondimensionalized into the curve of Fig (9), Ref
(8), which was originally intended for a design chart to aid in developing
strain response characteristics for use in fatigue. This curve displays a
parameter of response as the ordinate vs an excitation parameter on the
abscissa. Here, E is Young’s modulus, p is density, t is thickness, a is the
panel length of the short side, h is fluid depth, and the subscripts, p and F refer
to panel and fluid, respectively. A point not realized previously is that the
scaling shows that the thinner case can be used to represent the thicker panel
under the appropriate conditions and when nonlinearity is carefully
considered. More data with the thinner panels at the extreme conditions were
unfortunately not taken in several cases of strain response because of concern
with accumulating too many cycles before running the actual fatigue tests;
else the thinner cases could have shown even more dramatically the scale
effect.
Buffet has been of more interest in the past 15 years because of high angle of
attack operation of several modem USAF fighters. Much effort was placed
upon research with accurately scaled models to detennine if these could be
employed as in prior flutter work. The answer was YES! Several figures were
taken from Ref (16) to illustrate scaling of data from a model of , a fraction
of the size of a fighter, to the full scale quantity. Fig (10) shows the correlation
between scaled-up model data, flight test, and two sets of calculations over a
wide range of aircraft angle of attack for the F/A-18 stabilator. The data is for
inboard bending and torsion moment coefficients produced by buffeting
loads. The scaled model data correlates well, the calculations using Doublet
Lattice (DLM) aerodynamics is close, while the strip theory is not as accurate.
Fig (11) shows similar type of data for the F/A-18 Vertical Tail for outboard
bending moment coefficients. Here a wider range of angle of attack was
considered, and again scaled model data and calculations are close to aircraft
values. Both cases suggest that model data can be used to supplement full
scale work and that when combined with theory , are a powerful aid to full
scale analysis and tests. These tests can be used early in the aircraft design
cycle to insure full scale success.
CONCLUSIONS AND RECOMMENDATIONS
An attempt was made to employ a view of acoustical scaling different from
that usually taken. The idea is to develop data for a model that fits within a
test facility’s capability and then by using analytical methods, adjust these
results to the nominal case using factors from the test based on the ratio of
960
experimental to calculated data. This is analogous to the flutter model
approach. One example is offered, and similar results from related scaling in
fluid -structure and buffet work were shown to further the point. While more
work is needed to fully display the concept, enough has been done to inspire
others to dig-in and more fully evaluate the approach. The Writers intend to
do more research, since they fully appreciate this difficult task.
REFERENCES
1. Scanlon , R.H., and Rosenbaum, R., “ Introduction to The Study of Aircraft
Vibration and Flutter”, The MacMillian Company, New York, 1951
2. Bisplinghoff, R.L., Ashley , H. and , Halfman, R. L., “Aeroelasticity”,
Addision-Wesley Publishing Co., NY, Nov. 1955, pp. 695-787
3. Ferman, M. A., “Conceptual Flutter Analysis Techniques - Final Report”
Navy BuWeps Contract NO w 64-0298-c, McDonnell Report F322, 10 Feb.
1967
4. Ferman, M.A. and Unger, W. H., “Fluid-Structure Interaction Dynamics in
Fuel Cells”, 17th Aerospace Sciences Meeting, New Orleans, La. Jan 1979
5. Ferman, M.A. and Unger, W. H. , “Fluid-Structure Interaction Dynamics in
Aircraft Fuel Cells”, AlAA Journal of Aircraft, Dec. 1979
6. Ferman, M.A. , et al , “ Fuel Tank Durability with Fluid-Structure
Interaction Dynamics ,” USAF AFWAL TR-83-3066, Sept. 1982
7. Ferman, M..A., Unger, W. H., Saff, C.R., and Richardson, M.D., “A New
Approach to Durability Predictions For Fuel Tank Skins” , 26th SDM,
Orlando, FL, 15-17 April 1985
8. Ferman, M. A. , Unger, W. H., Saff, C.R., and Richardson, M.D. , “ A
New Approach to Durability Prediction For Fuel Tank Skins”, Journal of
Aircraft, Vol 23, No. 5, May 1986
9. Saff, C.R., and Ferman, M.A, “Fatigue Life Analysis of Fuel Tank Skins
Under Combined Loads”, ASTM Symposium of Fracture Mechanics ,
Charleston, SC, 21 March 1985
10. Ferman, M.A., Healey, M.D., Unger, W.H., and Richardson, M.D.,
“Durability Prediction of Parallel Fuel Tank Skins with Fluid-Structure
Interaction Dynamics”, 27th SDM, San Antonio, TX, 19-21 May 1986
11. Ferman, M.A, and Healey, M.D., “Analysis of Fuel Tank Dynamics for
Complex Configurations, AFWAL TR -87-3066, Wright-Patterson AFB, OH,
Nov 1987
12. Ferman, M.A., Healey, M.D. and Richardson, M.D.,” Durability Prediction
of Complex Panels With Fluid-Structure Interaction”, 29th SDM,
Williamsburg, VA, 18-20 April 1988
13. Ferman, M.A., Healey, M.D., and Richardson, M.D., “A Dynamicisf s
View of Fuel Tank Skin Durability, AGARD/NATO 68th SMP, Ottawa,
Canada, 23-28 April 1989
14. Zimmerman, N.H. and Ferman, M.A., “Prediction of Tail Buffet Loads
for Design Applications, USN Report, NADC 88043-30, July 1987
961
15. Zimmerman, N.H., Ferman, M.A., Yurkovich, R.N, “Prediction of Tail
Buffet Loads For Design Applications”, 30th SDM, Mobil, AL , 3-5 April
1989
16. Ferman, M.A., Patel, S.R., Zimmemian, N.H., and Gerstemkom, G., “ A
Unified Approach To Buffet Response Response of Fighters”, AGARD/NATO
70th SMP, Sorrento, Italy, 2-4 April 1990
17. Cote, M.J. et al, “Structural Design for Acoustic Fatigue”, USAF ASD-
TR-63-820, Oct 1963
18. Rudder, F.F., and Plumblee, H.E., Sonic Fatigue Guide for Military
Aircraft” AFFDL-TR-74-1 12, Wright-Patterson AFB, OH, May 1975
19. Jacobs, J.H., and Ferman, M.A. , Acoustic Fatigue Characteristics of
Advanced Materials and Structures, “ AGARD/NATO SMP , Lillehammer,
Norway, 4-6 May 1994
20. McDonnell Douglas Lab Report, Tech. Memo 253.4415, Acoustic Fatigue
Tests of Four Aluminum Panels, Two With Polyurethene Sprayon”, 27 June
1984
Figure 1 - Standard Acoustic Fatigue Design Method
962
MICROSTRAIN -
HYPOTHETICAL EXAMPLE
SOUND PRESSURE LEVEL -dB N- CYCLES TO FAILURE
Figure 2 - General Method for Estimating Fatigue Life at SPL
above Test Facility Capability
Figure 3 - Scaling Method Fits-in with General Design
Cycle for Acoustic Fatigue
963
SOUND PRESSURE LEVEL - dB
N@ cIBreq , Nom. Ext.
N @ dBj^ , Re-scaled
N @ dBj^ , Scaled Model
N - CYCLES TO FAILURE
Figure 4 - Hypothetical Examples of Acoustic Scaling
to Tests at Higher SPLs
HYPOTHETICAL CASE
Figure 5 - Statistical Aspects of Scaling
MICROSTRAIN -
^ MDC Tests
Ref. (19)
Figure 6 - Strain Response of an Aluminum Panel
(10 X 20 X 0.063 in. 7075 T6) Narrowband Random
Figure 7 - Example of scaled Model of the 10 x 20 in.
Aluminum Panel
965
Symbol Panel Fluid
Thickness Depth
(in.) (in.)
□
0.032
11.0
0
0.040
11.0
o
0.063
11.0
0
0.032
8.0
A
0.040
8.0
o
0.063
8.0
0
0.032
4.0
0.040
4.0
•
0.063
4.0
Figure 8 - Dynamic Strain vs Excitation Level - Sine
Figure 9 - Dynamic Strain Parameter vs Input Parameter
Mean Strain at Fluid Depth
966
0.04
Inboard Bonding
Inboard Torsion
0.032
RMS Buffet 0.024.
Moment
Coeffidant,
0.01 e
0.008
0 .
0 4 8 12 16 20 24
Angle of Attack - degrees
A
C
1
1
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1 0
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0 4 8 12 16 20 24
Angie of Attack - degrees
Figure 10 - F/A-18 Stabilator Buffet Correlation Study
0.048
0.04
0.032
RMS
Moment
CoefficiantO.024
BM/(QL3)
0.016
0.008.
0
16 24 32 40 48 56
Angle of Attack - degrees
Figure 11 - F- 18 Vertical Tail Buffet Response Moment Coefficients for
Angle-of-attack Variations
Outboard Bending Moment (70% Span, 45% Chord)
j— I j j
—
□ Ca
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968
ACOUSTIC FATIGUE II
THE DEVELOPMENT AND EVALUATION OF A NEW
MULTIMODAL ACOUSTIC FATIGUE DAMAGE MODEL
Howard R Wolfe
WL/FIBG Bidg24C
2145 Fifth St Ste2
Astronautics
Wright-Patterson AFB, OH
45433-7006
USA
Robert G. White
Head of Department
Department of Aeronautics and
University of Southampton
Southampton, S017 IBJ
UK
ABSTRACT
A multimodal fatigue model has been developed for flat beams and plates.
The model was compared with experimental bending resonant fatigue
lifetimes under random loading. The method was accurate in predicting
cantilevered beam fatigue lifetimes, but under predicted clamped- clamped
beam test results. For the clamped plate tests, one calculation was accurate
and the other predicted about half the test lifetimes. The comparisons and
the parameters affecting them are presented.
INTRODUCTION
While the single mode acoustic fatigue theory is satisfactory for sound
pressure levels around 158 dB overall and below, there is evidence in the
literature [1, 2, and 3] that above this level the accuracy of the simple
response prediction method decreases with increasing sound pressure
levels. The purpose of the paper is two fold, first to develop a multimodal
acoustic fatigue life prediction model ^d secondly, to evaluate its accuracy
in estimating the fatigue life theoretically by comparing predictions with
experimental results.
FATIGUE MODEL DEVELOPMENT
Many fatigue models are found in the literature. The Miner single mode
model used by Bennouna and White [4] and Rudder and Plumblee [5] was
selected to develop a multimodal nonlinear model. The fundamental
formulation is given by,
N.-fs^r «)
969
where N,. is the total number of cycles to failure, P(e) is the peak strain
probability density. N is the total number of cycles to failure at incremental
constant amplitude strain levels derived from a sinusoidal strain versus
cycles to failure curve. To calculate the fatigue life in hour, Eq (1) can be
expressed as,
t (hours) =
where t is time, Pp (sd) is the peak standard deviation probability density,
Nc is the total number of cycles to failure at a specified strain level and ^ is
the cyclic frequency. When the mean value is not zero, which is the case
with axial strain in the beam or plate, the rms value is not the standard
deviation. The standard deviation is usually employed to compute the time
to failure. Most of the S-N curves or e -N curves are approximated as a
straight line on a logarithmic graph. The relationship between the surface
strain and the cycles to failure is then.
8 =
(3)
where K is a constant and a the slope of a straight line on a log-log graph.
The cyclic frequency 4 for a single mode case is taken as the frequency of
the associated resonance. Two types of peak probability density techniques
were investigated from multimodal nonlinear strain responses [3]. These
were called major peaks and minor peaks. The major peaks were counted
for the largest peaks between zero crossings. The minor peaks were
counted for all stress reversals or a positive slope in the time history
followed by any negative slope. The effective cyclic frequency is much
lower for the major peak count than the minor peak count. However, the
peak probability density functions or PPDFs compared for these two cases
were almost the same. The major peak method was selected for further
study.
Given a particular peak probability density curve from a measured
response in an experiment, the number of peaks and the sampling time tg
can be used to determine the effective multimodal cyclic frequency,
fcm = number of peaks / tg (4)
970
where is the effective multimodal cyclic frequency. Substituting the
multimodal cyclic frequency into Eq (4),
t (hours) =
Pp(sd)
K/(e)‘/“
3600xfcm
(5)
This model accounts for the effects of axial strains which cause the mean
value not to be zero, nonlinear response and multimodal effects. If the
mean value is zero, then the standard deviation is equal to the rms value
and Eq (5) reduces to Eq (2).
FATIGUE MODEL COMPARISON WITH EXPERIMENTAL
RESULTS
The peak probability density function (PPDF) is needed or preferably
the time history from strain or dynamic response measurements to
evaluate the fatigue model developed. Also needed are sinusoidal 8 -N
curves for the structure, and knowledge of its boundary conditions and the
equivalent multimodal cyclic frequency.
Two types of peak probability density techniques were investigated from
multimodal nonlinear strain responses [3]. These were called major peaks
and minor peaks. The major peaks were counted for the largest peaks
between zero crossings. The nunor peaks were counted for all stress
reversals or a positive slope in the time history followed by any negative
slope. The effective cyclic frequency is much lower for the major peak coimt
than the nunor peak coimt However, the peak probability density
functions or PPDFs compared for these two cases were almost the same.
The major peak method was selected for further study.
Comparison with Beam Data:
The K and a terms were calculated from 8-N and S-N data, where S
is stress, using Eq (3). Selecting two values of strain and their
corresponding cycles to failure, yields two simultaneous equations
which were solved for K and a. Two sinusoidal E-N curves for
BS1470-NS3 aluminum alloy which has a relatively low tensile strength
were obtained from Bennouna and White [4 Fig 8]. These were for a
cantilevered beam and a clamped-clamped (C-C) beam as shown in Fig
1. The K and a terms calculated were used to compute the cycles to
failure, Nc, for each strain level. Table I shown is the same as Table I in
reference [4] except was calculated from Eq (3) to sum the damages.
971
Delta is the sample size. The cyclic frequency was for one mode the one
resonant response frequency. The time to failure in hours using Eq (2)
for the cantilevered beam was 16.6 hours compared with 16.2 predicted
theoretically [4], 15.3 and 15.9 obtained experimentally [4]. Both
theoretical results were essentially equal, but slightly higher than the
test results. The time to failure from Eq (2) for the (C-C) beam was 3.04
hours compared with 2.53 theoretically [4] and 5.25 and 5.92
experimentdly [4]. Both theoretical results were about one half of the
test results. The K and a terms, the theoretical fatigue life times and the
experimental fatigue life times are listed in Table H. The table contains
three sections: lifetimes calculated using a strain gauge PPDF, a
displacement PPDF and the Gaussian and Rayleigh PDFs. As noted in
reference [4], failure occurred much earlier for the C-C beam than the
cantilevered beam for the same strain level. This was attributed to the
influence of a large axial strain in the clamped- clamped beam.
Comparison with Plate Data:
Two fatigue tests were conducted to provide some additional limited
data for comparison with the fatigue model developed. These tests
used the base excitation method with a 1.09x10® N (20,000 Ibf)
electrod)mamic shaker. The clamping fixture consisted of a flat
aluminum alloy 6061-T6 plate 19 mm thick and four clamping bars of
equal thickness. The radius of curvature of the clamping edges was 4.76
mm to prevent early fatigue failure. A four bar clamping arrangement
was selected to prevent buckling of the plate while torquing the
clamping bolts. The undamped size was 254 x 203 x 1.30 mm which
results in a 1.25 aspect ratio. Strain gauges were bonded along the center of
the larger dimension (SG 2) and at the center of the plate (SG 3).
Displacements were measured with a scanning laser vibrometer at the
center of the plate. An accelerometer was moimted on the shaker head to
determine the acceleration imparted to the damped plate. A flat
acceleration spectral density was used between 100-1500 Hz. Recordings
were taken at increasing levels of exdtation up to the fatigue test level. The
time to detecting the first fatigue crack was recorded for each plate.
The constants K and a were calculated from random single mode S-N
data for 7075-T6 aluminum alloy [5 p 489] shown in Fig 1, with
K = 1.01x10^^ and a = - 0.175. The rms stress ( Srms ) ^^s changed to rms
strain, Srms = EBrms, where E is Young's modulus. The stress was
measured half way between two rivets along the center line between the
rivets on the test specimen. The strain gauge location, stress
concentrations, and the boundary conditions greatly affect the strain
972
level measured. Correction factors are needed for a different set of
conditions and to convert random data to sinusoidal data. Sinusoidal
£ -N bending coupon curves for 7075-T6 aluminum alloy were difficult to
find. S-N curves were found for an aerospace material with both
sinusoidal and random excitations. These curves were nearly parallel.
The sinusoidal strain was 1.38 times larger than the random strain for
10^ cycles. Multiplying the constant K for the 7075-T6 material by 1.38
resulted in K = 1.40x10^^ .
Early strain gauge failures prevented strain measurement above 500
microstrain with 20.7 g rms shaker excitation. The fatigue test level was
115 g rms and the response contained at least six frequency response
peaks. The major peak strain PPDFs were determined for 5.32 g rms
and 20.7 g rms as shown in Fig 2 with the Gaussian PDF. Compared
with the Gaussian distribution, an increased number of peaks occurred
greater than 1 sigma and smaller than -1 sigma. Also a larger number of
peaks occurred around zero. The PPDF determined from the 20.7g rms
test case was used to predict fatigue life, but a new strain estimate was
needed since the excitation level increased 5.6 times. The displacement
is directly related to the strain at each excitation level. Displacement
measurements at the fatigue test level were used to estimate the strain
level shown in Fig 3. The estimated strain from the figure was 770
microstrain for SG 2. The scale of the displacement measurements was
adjusted to coincide with strain measurements at increasing increments
of shaker excitation.
The equivalent cyclic multimodal frequency is needed to predict the
fatigue life. Prediction of the linear modal frequencies is carried out by
a variety of methods. Usually the first mode prediction is the most
accurate. The cyclic multimodal nonlinear frequencies have been
studied for two clamped beams and two clamped plates [3]. These were
based upon the peak probability density functions (PPDFs) where the
peaks were counted for a specific time interval, from which the
nonlinear cyclic multimodal frequencies were calculated. Generally the
resonant frequencies increased with increasing excitation levels. Those
for the two beams increased more rapidly than those for the two plates.
Very little change was noted for the plates. The equivalent cyclic
miiltimodal frequency determined via Eq (4) from the SG 2 PPDF was
348 Hz.
The time to failure in hours predicted using Eq (5) and the parameter
mentioned above for the clamped plate was 0.706 hours compared with
1.17 and 0.92 shown in Table 11. The predicted result was slightly lower
973
than the test results. A Srms’^ curve was calculated with the test
lifetimes available, by determining a new constant K, assuming the
slope was the same as for the riveted coupon and applying the
sinusoidal correction factor. The time to failure in hours predicted
using Eq 5 and the calculated e-N curve for the clamped plate was
0.274 hours compared with 1.17 and 0.92 shown in Table E. The cyclic
multimodal frequency used was the same at that determined from SG 2
and the same strain was used. This prediction was about 1/3 of the test
results. This method incorporates the failure data at two points.
The displacement PPDF shown in Fig 4 was used to predict fatigue
life. The number of displacement peaks increased significantly above
the strain PPDF around 1 sigma and -1 sigma. The large number of
peaks around zero was similar to the characteristics observed in the
strain PPDF. The equivalent cyclic multimodal frequency was 375 Hz,
slightly higher than that determined from the strain PPDF. However,
the same frequency (348 Hz) was used to predict fatigue life. The time to
failure in hours using Eq (5) for the clamped plate was 1.15 hours with
the riveted e-N curve and 0.446 with the calculated 8-N curve shown
in Table E. This PPDF improved considerably the prediction. The
Gaussian and Rayleigh PDFs were used to predict the time to failure
with the same parameters as those used with the riveted e-N curve.
The lifetime using the displacement PPDF was 1.15 hours, using the
Gaussian PDF, 0.600 hours and using the Rayleigh PDF, 0.237 hours, as
shown in Table E. The Gaussian PDF under predicts by a factor of 2.
The Rayleigh PDF under predicts by a factor of 5.
The spread sheets containing PPDF / Ncm data for various sigma
values were used to determine damage accumulation shown in Fig 5.
Almost 55% of the normalized damage occurs between -2 and -1 sigma
and 38% of the damage between 1 and 2 sigma using the displacement
PPDF. However, the damage is spread more evenly using the available
strain gauge PPDF. The strain gauge PPDF was recorded at a much
lower level than the displacement PPDF. The damage accumulation
compared more closely to Gaussian PDF than the Rayleigh function.
Damage Model with a Specific Function Describing the PPDF:
A curve-fitting routine was used to determine a mathematical function
for a high level strain gauge PPDF for the clamped shaker plate. The
most important part of the fit is outside the range of -1 to 1 sigma, since
most of the damage accumulation occurs outside this range. The
highest ranking function was a tenth order polynomial followed by
974
ninth and eighth order polynomial fits. The goodness of fit in order
from 1 to 14 ranges from 0.9775 to 0.9625, which are very close
statistically. The tenth order polynonual is,
y = a 4* bx + cx^ + dx^ + ex"^ + fx^ + gx^ + hx^ + ix^ + jx^ + kx^^ (6)
where a=0.346, b=-0.0148, c=-0.137, d=-0.054, e=0.090, f=0.043, g=-
0.0400, h=-0.00976, i= 0.00722, j=6.85xl0"‘, k=<4.36xl0'" . Ranked
fourteenth is a natural logarithmic function. The function and its
coefficients are,
Iny = a + bx + cx^ + dx^ +ex^ 4-fx^ (7)
where a=-1.088, b=-0.1191, c=-0.1.302, d=0.0104, e=-0.0653, and f=0.0079.
The function fits the test data similar to the tenth order polynomial and
may be easier to use. Ranked forty-first is a Gaussian function. The
function and its coefficients are.
y = a + b exp|o.5[(x - c) / d]^ |
(8)
where a=-0.0968, b=0.4485, c=-0.050 and d=1.45. The function fits better
for sigma values of 2 or greater than those of -2 sigma and greater. A
constant coefficient is used to fit the Gaussian function to permit shifting
the function to fit the test data. This equation can be used in the PPDF
in Eq (5),
a + b exp [-0.5 [(x - c) / d]^ }
- [K/(e)f“
3600£
■cm
(9)
where a=-0.0968, b=0.4485, c=-0.050 and d=1.45.
CONCLUSIONS
The prediction of multimodal fatigue life is primarily dependent upon
the peak probability density function (PPDF) which changes shape with
increasing excitation levels. The next in order of importance is the
sinusoidal e -N bending fatigue curve and finally the effective multimodal
cydic frequency.
A multimodal fatigue model was developed with the PPDF estimated
from a form of the Gaussian function being useful especially in the
975
range of cydes to failure from 10^-10^. The lifetime predication
calailations for the damped-clamed beam was about one half the
experimental value. For the plate, the calculations was about one half
the experimental value. Using riveted coupon fatigue data, the
calculation was accurate.
REFERENCES
1. B.L. Clarkson, April 1994, '‘Review of sonic fatigue technolog/',
NASA contractor report 4587, NASA Langley Research Center,
Hampton, Virginia.
2. R.G. White, October 1978, "A comparison of some statistical
properties of the responses of aluminium alloy and CFRP plates to
acoustic excitation". Composites 9(4), 125-258.
3. H.F. Wolfe, October 1995, "An experimental investigation of
nonlinear behaviour of beams and plates excited to high levels of
dynamic response", PhD Thesis, University of Southampton.
4. M. M. Bennouna, and R. G. White, 1984, "The effects of large
vibration amplitudes on the dynamic strain response of a clamped-
clamped beam with consideration on fatigue life". Journal of Sound and
Vibration, 96 (3), 281-308.
5. J. R. Ballentine, F. F. Rudder, J. T. Mathis and H.E. Plumblee,
1968, 'Refinement of sonic fatigue structural design criteria", AFFDL TR
67-156, AD831118, Wright-Patterson AFB, Ohio.
TABLE I
FATIGUE CALCULATIONS USING EQUATION 5.9
CANTILEVERED BEAM BS 1470-NS3 (REF 4 FIG 8)
e/sd
sd
P£
e
ll£
Nc=(2.172
xl0“/sd)“““
PPDF X A
PPDF X A/Nc
t
hours
0.5
H
213
15079045.41
n
1.41256E-07
1
|g
425
4390936.857
mam
7.06000E-07
1.5
425
638
2125446.064
WBrn
1.12917E-06
2
425
850
1273261.536
0.138
1.08383E-06
2.5
425
1063
854026.6252
0.060
7.02554E-07
3
425
1275
617189.6449
0.018
2.91645E-07
3.5
TOTAL
425
1488
468372.0662
0.005
0.984
1.06753E-07
4.16121E-06
16.7
976
TABLE n
SUMMARY OF FATIGUE CALCULATIONS
Figure 1 8 -N curves for aluminum alloys tested.
Figure 2 Normalized strain PPDF comparison with a Gaussian PDF.
978
NORMALIZED DISPLACEMENT
MOJORPPDF
2
Figure 4 Normalized displacement PPDF comparisons,
9
980
ACOUSTIC FATIGUE AND DAMPING TECHNOLOGY IN
COMPOSITE MATERIALS
By B. Benchekchou and R.G. White
Abstract
Considerable interest is being shown in the use of composite materials in
aerospace structures. Important areas include development of a stiff, lightweight
composite material with a highly damped, high temperature polymer matrix
material. The study described in this paper concerns the application of such
material in the form used in thin skin panels of aircraft and investigation of its
fatigue properties at room and high temperature. For this purpose, flexural fatigue
tests have been carried out at two different temperatures and harmonic
three-dimensional FE analyses were performed in order to understand the
dynamic behaviour of plates. Random acoustic excitation tests using a
progressive wave tube, up to an overall sound pressure level of 162 dB, at room
temperature and high temperatures were also performed in order to investigate the
dynamic behaviour of panels made of the materials. Various methods for
including damping in the structure were examined when parameter studies were
carried out, and conclusions have been drawn concerning optimal incorporation
of a highly damped matrix material into a high performance structure.
1-Introduction
Significant areas of primary and secondary structures in military aircraft operate
at high temperature and are subjected to high levels of random acoustic loading,
because of their closeness to jet effluxes. There is then a need to develop a carbon
fibre reinforced plastic material with a high temperature polymer matrix and high
fatigue resistance. Highly damped composite structures should be developed in
order to better resist dynamic loading and to have an enhanced fatigue life. Work
previously carried out on improving the damping in fibre reinforced plastic (FRP)
composites as well as the number of approaches which can be taken to improve
the damping properties of polymeric composites have been summarised in [1].
The aim of the research described here was to study lightweight composite
materials with a highly damped, high temperature polymer matrix material, by
981
investigating its mechanical and acoustic fatigue properties, the latter investigation
being carried out using thin, multilayered plates.
2-ExperimentaI work
For this type of study, two adequate prepregs were highlighted after investigation:
SE300 and PMR15. The SE300 material was carbon fibre reinforced prepreg of
(0°/90°) woven form, 0.25 mm thick and had 60% fibre volume fraction, with no
suitable data available on the material properties. Dynamic mechanical thermal
analyser (DMTA) analyses carried out on four specimens ( 20 nun long and 12
mm wide) with different lay-ups i.e. (0°/90°)4, (+45°/-45°)4, (0°/90°, 45°/45°)s
and (45®/45°;0®/90°)s, allowed to get provided the material properties. Results
from DMTA analyses are shown in Table l.a where the loss factor and the
Young’s modulus values at 40^C and at the glass transition temperature Tg are
presented. The loss factor values varied from 0.0097 to 0.085 for a range of
temperature from 40®C to 300®C.
The PMR15 prepreg was also of (0^/90^) woven carbon form and had 58%
volume fraction. Six DMTA specimens having the following lay-ups: (0®/90®)4,
(00/90^)8, (4450/-450)4, (+450/-45<^)8, (0^/90^-, 450/450)s and
(450/450.00/900)^ were made. Results from DMTA analyses showed that the loss
factor values varied from 0.0129 to 0.0857 for a range of temperature from 40®C
to 400®C, with a value of loss factor of 0.1293 at 375®C, the maximum
temperature for normal use being 352^C. The loss factor and Young’s modulus
values at 40®C and Tg are given in Table l.b.
Mechanical behaviour of the selected materials
The fatigue characteristics of these new materials were investigated and results
were compared with those of well established structural materials. Mechanical
fatigue tests of SE300 and PMR15 samples using "sinusoidal" loading at a chosen
maximum strain level, i.e. 8000 pS were carried out and performances compared
to that of an XAS/914 sample. A mechanical (flexural) fatigue rig was used for
this purpose to test specimens in a cantilevered configuration. Details of the rig are
available in [2]. The particular clamp used was designed by Drew [3] to induce
damage in the centre of the specimen instead of having edge damage, i.e. peeling
while flexural tests are carried out. In order to investigate the performance of these
new materials at high temperature, fatigue tests were also carried out on samples,
at 210^C . This was achieved by using a heating system which consisted of two air
982
blowers (electronically regulated hot-air guns) positioned at 40 mm above and
below the specimens, which allowed specimens to be tested at a uniform temperature
of 210±5°C. The aim of the mechanical fatigue test was to determine the number of
loading cycles needed for damage to occur and its subsequent growth rate in
cyclically loaded composite specimens of SE300 and PMR15 matrix materials . The
samples were 140 mm long, 70 mm wide and 2 mm thick. Fatigue tests of SE300 (S3
and S4) and PMR15 (PI and P2) specimens, at room temperature and at 210OC
respectively, at a level of 8000 p.S, located by the peak of the half-sine clamp, have
been carried out. Ultrasonic scans of specimens S3 and S4 before any loading cycles
and after 100, 500, 1000, 2000, 5000, 10000, 20000 and 50000 loading cycles are
shown respectively in Figures l.a-h and Figures 2.a-h. A small delamination,
indicated by lighter areas in the scans, starts to show in both specimens S3 and S4
after applying 500 loading cycles and increases substantially after 5000 loading
cycles. After 5000 loading cycles, the damage area increased more for specimen S4
than specimen S3, which shows that the latter is slightly more fatigue resistant. In
other words, when increasing the temperature from 250C to 210<^C, the resistance to
fatigue slightly decreases. Figures 3.a-h and Figures 4.a-h show the ultrasonic scans
for PMR15 specimens PI and P2 before and after several loading cycles. For both
specimens PI and P2, damage starts after 500 loading cycles and increases
substantially after 20000 loading cycles. At this stage, delamination areas are similar
for both specimens PI and P2 and just a little more pronounced in specimen P2,
which shows that the latter is slightly less fatigue resistant. Hence, an increase in
temperature leads to a decrease in the fatigue resistance properties of the specimens.
From Figures 1, 2, 3 and 4, one can conclude that the PMR15 specimens tested were
slightly more fatigue resistant than the SE300 specimens. In fact, damage in
specimens was generally more defined, clearer and spread more rapidly in the SE300
samples than was the case for the PMR15 samples. Figure 5 shows damage
propagation occuring in an XAS/914 sample (XI), with (0o/±45®/0O)s stacking
sequence, tested at 8000 |iS level and at room temperature, from [3]. Note that
substantial damage existed after 1000 loading cycles in this specimen, which shows
that both SE300 and PMR15 are more fatigue resistant than XAS/914 at room
temperature.
Acoustic fatigue behaviour of panels of the selected materials
Investigations were carried out by installing the CFRP plates in an acoustic
progressive wave tunnel, (APWT) in order to determine the response of CFRP plates
under broadband acoustic excitation simulating jet noise. The plate was fully
clamped around its boundaries on to a vertical steel frame fixed to one side of the
APWT, so that it formed one of the vertical walls of the test section of the APWT.
983
Overall sound pressure levels (OSPL) up to 165 dB of broadband noise in the test
section of the tunnel was generated by a Wyle Laboratories WAS 3000 siren. A
heater panel capable of heating and maintaining the temperature of test plates up to
300*^0 while mounted on the tunnel was designed and built. Temperatures were
monitored and controlled via thermocouples on the panel. Plates were excited by
broadband excitation in the frequency range 80-800 Hz. A B&K type 4136
microphone mounted at the centre of the test section of the tunnel adjacent to the
mid-point of the plate were used for sound pressure measurements. Eight strain
gauges, four on each side of the plate were attached in order to monitor the strain
distribution in the panel while the rig was running; more details of the experiment
may be found in [1]. Acoustic tests were run, at various temperatures and OSPL.
Since plates were excited in the frequency range 80-800 Hz, spectral analyses
would not include the first natural frequency. The natural fundamental frequency of
an SE300 clamped panel was found to be 49.02 Hz analytically. The second -and
third resonance frequencies were 149.5 and 198.5 Hz, as calculated from strain
spectral densities, from tests carried out with an OSPL of 156 dB and a temperature
of 1620C. At 162 dB, results showed that the second resonance frequency was 113
Hz at 150OC and 106.5 Hz at 195^0, which shows that when the temperature and
the OSPL increased, the resonance frequencies of the plates decreased. Also, it was
found that the damping increased at elevated temperatures. The overall modal
viscous damping ratios, for the second mode, were calculated from strain spectral
densities, for an SE300 panel driven at an OSPL of 162 dB and at 1950C, and was
found to be 8.91%; this value is similar to that calculated from analytical
simulations, for the first mode, which is 8,50% at 242^0, (see the analytical section
below). A typical strain spectral density obtained from recorded results is presented
in Figure 6 for an SE300 specimen, at an OSPL = 156 dB and at 1620C, from a
strain gauge in the centre of the specimen. Maximum RMS strain values recorded
from experimental tests, at a strain gauge in the centre of the specimen were, at an
OPSL =156 dB, 1300 \iS, 1800 \iS and 2800 pS at room temperature and at 90^0
and 1620C respectively. These results clearly indicate a trend for significant
increase in dynamic response with increasing temperature.
Experiments were also carried out on a PMR15 panel at various OSPL and
temperatures. Results from tests run at 159 dB and at room temperature show that
the second and third resonance frequencies were indicated as 112 and 182.5 Hz.
When the OSPL increased to 162 dB, the second and third resonance frequencies
decreased to 110.5 Hz and 176.5 Hz. At the same OSPL (162dB) and when
temperature increased to 2810C, the third resonance frequency became 139 Hz. This
984
shows that, for PMR15 plates, when the temperature and the OSPL increased, the
resonance frequencies of the plates decreased. It was also clear that modal damping
increased with increasing temperature. In fact, the overall viscous damping reached
20%, for the second mode, at an OSPL=162 dB and at 2810C. It must be stated here,
however, that apparent damping trends could include nonlinear effects which
influence bandwidths of resonances. Maximum RMS strain values recorded at room
temperature, by a strain gauge in the centre of the plate were found to be: 2700 pS at
153 dB, 2800 pS at 157.9 dB and 2900 pS at 159 dB.
It is clear from these values that increasing the OSPL obviously leads to an increase
of the strain in the plate. Similar results were observed when the temperature was
increased. In fact, at an OSPL of 162 dB, the maximum strain values recorded by a
strain gauge in the centre of the specimen were 3000 pS, 3400 pS and 5000 pS at
105^0, 1650C and 281^0 respectively, which clearly indicates the effects of
temperature. It was observed that both the PMR15 and SE300 panels behaved in a
non linear manner.
Attempts to acoustically fatigue a PMR15 panel were made at 162 dB. No signs of
fatigue damage were shown in an ultrasonic scan of the panel after 1389 minutes of
running time.
3-Analytical work
In order to examine various methods for including damping in a structure, parametric
studies were carried out using the finite element FE method. ANSYS software has
been used. A three-dimensional, 3D layered element, SOLID46 was used to build
theoretical models. The element is defined by eight nodal points, average layer
thickness, layer material direction angles and orthotropic material properties, [4].
Meshes were built in order to carry out modal and harmonic analyses of multilayered
composite plates (410 mm, 280 mm, 2 mm). The plates were fully clamped along all
edges, in order to simulate the panels tested in the APWT. Natural frequencies were
first determined from free vibration analyses and compared to resonance frequency
values derived from experimental data. Then, the plate was driven by harmonic
loading at one point of application. The forcing frequency varied from 0 to 400 Hz.
The amplitude of the load was 50 N. Results for displacements and response phase
angles relative to the force for a chosen position on the plate as a function of
frequency were obtained. The approach was then to carry out parameter studies in
order to examine various methods for including damping in the structure, i.e. to use
highly damped matrix material throughout the whole structure or possible
985
incorporation in a few layers. Structural damping was included, allowing models to
run with different damping values in each ply of the panel. Structural damping is
inherent in the structure and depends on the natural frequency; details on structural
damping modeling may be found in [1]. Analyses were performed considering
structural damping for the first mode. The structural damping was then varied for
plies with the same orientation for a viscous damping ratio ^ = 0.01, 0.02, 0.05, 0.10
and 0.20.
Simulations with SE300
Models were built up with the following stacking sequence ((45°/45°),(0°/90°))s,
lay-up used for the experimental plates. Table 2 gives the first three modal
frequencies of the panel obtained from free vibration analyses results. Harmonic
simulations were carried out and the overall damping value was calculated for each
case with results given in Table 3. As can be seen, if high overall damping is needed
for a structure composed of the SE300 material, increasing the damping value of the
(45°/45®) orientation plies most significantly increases the overall damping value of
the panel. In fact, putting a damping value of 20% in the (45°/45°) orientation plies
leads to an overall viscous damping value of 14.52%, which is better than including
a 10% damping value in all of the plies of the structure.
Harmonic analyses of fully clamped plates were also performed with the values of
material properties taken at several temperatures. Simulations were carried out with
material properties at 2420C and 300OC. Free vibration analyses permitted
calculation of the modal frequencies of the panels at the temperatures mentioned
above. Table 2 also lists the first three modal frequencies from analyses with
material properties at 242^0 and 300^0. The overall viscous damping values,
obtained from FE simulations, are given for each ternperamre in Table 4. Again, the
damping value has been varied through the layers and the overall damping value was
calculated in order to see which of the plies contributes the most to heavily damp the
plate. It was found that putting a damping value of 20% in the (45°/45°) orientation
plies, the first mode viscous damping ratios were 14.62% and 14.55% at 242^0 and
300OC respectively. This shows that this material is more highly damped at high
temperature and presents better damping properties of the two materials at 242^0.
Simulations with PMR15
Free vibration analyses of models built up with the following stacking sequence
((45°/45°),(0°/90®))s were carried out and the first three modal frequencies of the
panel are shown in Table 5. Harmonic analyses were run and the overall damping
986
value was calculated for each simulation with results given in Table 6. If high
overall damping is needed for a structure composed of the PMR15 material,
increasing the damping value of the (45°/45®) orientation plies most significantly
increases the overall damping value of the panel. In fact, putting a damping value
of 20% in the (45°/45°) orientation plies leads to an overall viscous damping value
of 14.39%, while if (0°/90®) orientation plies have a 20% damping value, the
overall damping is 7.42%.
Harmonic analyses of fully clamped plates were also carried out with the values of
material properties taken at several temperatures. Simulations were carried out
using material properties at 3750C and 400oC. Free vibration analyses permitted
calculation of the modal frequencies of the panels at the temperatures mentioned
above. Table 5 lists the first three modal frequencies from analyses with material
properties at 3750C and 400^0. The overall viscous damping values, obtained
from FE simulations, are given for each temperature in Table 7. Again, the
damping value has been varied through the layers and the overall damping value
was calculated in order to see which of the plies contributes the most to heavily
damp the plate. It was found that putting a damping value of 20% in the (45‘^/45°)
orientation plies, the first mode viscous damping ratios were 18.39% and 16.94%
at 3750c and 400OC respectively. This shows that this material is more highly
damped at high temperature and presents better damping properties of the two
materials at 3750C.
4~Conclusions
Two matrix materials, SE300 and PMR15, with potential for use in aircraft
structures in a severe environment, i.e. temperatures up to SOO^C were selected for
this study. Material properties were determined using DMTA techniques and
results show that these materials have high damping abilities at high temperature.
Dynamic loading tests, performed in flexure at room and high temperature showed
that the carbon fibre reinforced PMR15 material is more fatigue resistant than
SE300 and XAS/914 based composites. Acoustic tests using a progressive wave
tunnel, up to a random acoustic OSPL of 162 dB, at room temperature and
elevated temperamres up to 2810C were also performed. When increasing the
excitation level and the temperature higher strain values in the centre of the panels
were recorded. Free vibration and harmonic FE analyses permitted determination
of the natural frequencies and the overall viscous damping values. Resonance
frequencies determined from results obtained from acoustic tests were similar to
987
natural frequencies obtained from FE simulations. Overall viscous damping values
obtained from experimental results agreed well with those obtained from the FE
analyses for SE300 panels. Results obtained for PMR15 panels, from tests, were
higher than those calculated analytically. Both tests and simulations showed that
SE300 and PMR15 present higher damping capabilies at high temperatures.
Conclusions, via parameter studies including material damping, have been drawn
concerning optimal incorporation of a highly damped matrix material into a high
performance structure.
5-AcknowIegments
The authors wish to thank the Minister of Defence for sponsorship of the programme
of research under which the work was carried out. Thanks are also due to Dr M. Nash
of the DRA, Famborough for many helpful discussions throughout the project. _
6-References
1- Benchekchou, B. and White, R.G., Acoustic fatigue and damping technology in
FRP composites, submitted to Composite Structures.
2- Benchekchou, B. and White, R.G., Stresses around fasteners in composite
structures in flexure and effects on fatigue damage initiation: I-Cheese-head bolts.
Composite structures, 33(2), pp. 95-108, November 1995.
3- Drew, R.C. and White, R.G., An experimental investigation into damage
propagation and its effects upon dynamic properties in CFRP composite material .
Proceedings of the Fourth International Conference on Composite Structures, Paisley
College of Technology, July 1987.
4- ANSYS theoretical manual, Swanson Analysis Systems Inc, December 1992.
988
Table La: Loss factor and Young’s modulus values at 40^C and at Tg for
SE300 samples analysed by the DMTA.
stacking sequences
(4.45/-45)4
(-457+45)4
(45/45;0/90)s
(0/90)4
Tg(°C)
242
242
240.3
238.71
T| atTg
0.085
0.085
0.081
0.061
Ti at 40°C
0.012
0.014
0.010
0.0097
Log E’ at40°C
9.870
9.840
9.970
10:097
Table Lb: Loss factor and Young’s modulus values at 40^C and at Tg for
PMR15 samples analysed by the DMTA.
Stacking sequences
(+45/--45)4
(0/90;45/45)s
(45/45;0/90)s
(0/90)4
Tg (OC)
372
375
375
375
Tl atTg
0.117
0.124
0.121
0.129
T| at 40®C
0.0110
0.0138
0.0086
0.0117
Log E’ at 40OC
9.583
9.944
9.972
9.875
989
Table 2: The first three modal frequencies for SE300 panel; analyses carried out
with material propert ies at room temperature, at 242*^0 and at 300°C.
Room temperature
242^0
300OC
49.02Hz
44.72Hz
43.61Hz
155.80Hz
143.27Hz
140.17Hz
212.04Hz
194.19Hz
189.68Hz
Table 3: Overall viscous damping values of SE300 panel. Values are calculated from
results obtained from harmonic analyses; the material damping being considered for
the first mode.
Simulation with
damping of
(45°/45°)orientation
plies
(0®/90°)orientation
plies
5%
3.81%
2.51%
10%
7.06%
3.92%
20%
14.52%
11.57%
Table 4: Overall viscous damping values of SE300 panel, for the first mode.
Temperature
(°C)
Overall viscous damping
25
1.20%
242
8.50%
300
5.45%
990
Table 5: The first three modal frequencies for PMR15 panel; FE analyses carried out
with material properties at room temperature, 375°C and 400°C.
Room temperature
3750c
400OC
43.43Hz
32.68Hz
25.54Hz
130.35Hz
99.65Hz
78.83Hz
183.96Hz
139.04Hz
109.11Hz
Table 6; Overall viscous damping values of PMR15 panel. Values are calculated
from results obtained from harmonic analyses; the material damping being
considered for the first mode..
Simulation with
damping of
(45°/450)orientation
plies
(0o/90O)orientation
plies
5%
3.76%
2.44%
10%
7.26%
4.10%
20%
14.39%
7.42%
Table 7: Overall viscous damping values of PMR15 panel, for the first mode..
Temperature
Overall viscous damping
(OQ
25
1.33%
275
13.24%
400
8.6%
991
a: before any loading cycles b: after 100 loading cycles c: after 500 loading cycles
d: after 1000 loading cycles e: after 2000 loading cycles f: after 5000 loading cycles
g: after 1 0000 loading cycles h: after 20000 loading cycles
Figure 1. Ultrasonic scans of specimen S3 after applying different numbers of loading cycles.
(SE 300 material, ambient temperature)
a: before any loading cycles b: after 100 loading cycles c: after 500 loading cycles
d: after 1000 loading cycles e: after 2000 loading cycles f: after 5000 loading cycles
Fisure 2. Ultrasonic scans of specimen S4 after applying different numbers of loading cycles.
(SE300 material, 210OC)
992
a: before any loading cycles b: after 100 loading cycles c: after 500 loading cycles
d: after 2000 loading cycles e: after 20000 loading cycles f: after 50000 loading cycles
g: after 100000 loading cycles
Figure 3. Ultrasonic scans of specimen PI after applying different numbers of loading cycles
(PMR15 material, ambient temperature, 8000]LlS)
a: before any loading cycles b: after 500 loading cycles c: after 1000 loading cycles
g: after 10000 loading cycles
h: after 20000 loading cycles
Figure 4. Ultrasonic scans of specimen P2 after applying different numbers of loading cycles.
(PMR15 material, 2100C, 8000M.S)
993
a: before any b: after 100 c: after 500 d: after 1000
loading cycles loading cycles loading cycles loading cycles
e: after 2000 f: after 5000 g: after 10000 h: after 20000
loading cycles loading cycles loading cycles loading cycles
Figure 5. Ultrasonic scans of an X AS/9 14 specimen fatigued at a level of 8000 llS
showing the damage propagation; the lay-up is (0/±45/90)s, [3],
0 Lin Hz RCLD 1.6k
Figure 6: SE300 specimen SI strain spectral density, recorded from strain gauge
ST2, OSPL =156 dB, temperature = 162^C.
994
THE BEHAVIOUR OF LIGHT WEIGHT HONEYCOMB SANDWICH
PANELS UNDER ACOUSTIC LOADING
David Millar
Senior Stress Engineer
Short Bros. PLC
Airport Road
Belfast
Northern Ireland
SUMMARY
This paper discusses the results of a progressive wave tube test on a carbon composite
honeycomb sandwich panel. A comparison was made with the test panel failure and the
failure of panels of similar construction used in the intake ducts of jet engine nacelles.
The measured panel response is compared with traditional analytical methods and finite
element techniques.
Nomenclature
= Overall rms stress (psi) or strain (jxs).
7t =3.14159
= Fundamental frequency (Hz).
5 = Critical damping ratio (*0.017).
Lps(fn) = Spectrum level of acoustic pressure (- expressed as a fluctuating rms pressure in psi in a
1 Hz band).
jr = Joint acceptance function (non dimensional).
= Characteristic modal pressure (psi)
ph = Mass per unit area (Ib/in^)
Sic “ Modal stress (psi) or modal strain (|j£).
Wjc = Modal displacement (in),
a = Panel length (in),
b = Panel width (in).
x,y,z = Co-ordinate axes.
1.0 Introduction
Honeycomb sandwich panels have been used for some time in the aircraft industry as
structural members which offer a high bending stiffness relative to their weight. In
particular, they have proved very attractive in the construction of jet engine nacelle
intake ducts where, in addition to their load carrying ability, they have been used for
noise attenuation.
995
2.0 Acoustic Fatigue
The intake duct of a jet engine nacelle can experience a severe acoustic environment
and as such the integrity of the nacelle must be assessed with regard to acoustic fatigue
[1], Acoustic fatigue characterises the behaviour of structures subject to acoustic
loading, in which the fluctuating sound pressure levels can lead to a fatigue failure of
the structure. The traditional approach to acoustic fatigue analysis has assumed
fundamental mode response and given that aircraft panels will in general, have
fundamental frequencies of the order of several hundred hertz, it is clear that the
potential to accumulate several thousand fatigue cycles per flight can exist.
Techniques for analysing the response of structures to acoustic loads were developed
originally by Miles [2] and Powell [3], Other significant contributions are listed in
References 4-7. Design guides such as AGARD [8] and the Engineering Sciences
Data Unit (ESDU) series of data sheets on vibration and acoustic fatigue [9], have
proved useful in the early stages of design.
Note - further details on the general subject of acoustic fatigue can be found in Ref
10, while a more detailed review of the subject up to more recent times is presented in
Ref 11.
3.0 In Service Failures
In recent years a number of failures have been experienced involving intake barrel
honeycomb sandwich panels. Failures have been experienced with panels which had
both aluminium facing and backing skins and carbon composite panels. The metal
intake liner was observed to have skin cracking and also core failure, while the
composite panel was only observed to have core failure.
With regard to the metal panels, flight testing was carried out and the predominant
response frequency was observed to be at the fan blade passing frequency - much
higher than the fundamental frequency of the intake barrel; this went some way to
explaining why the traditional approach in estimating the response did not indicate a
cause for concern. The response of the panel was also very narrow band - almost a
pure sinusoid (again differing from the traditional approach of broad band/random load
and response), and the subsequent analysis of the results was based on a mechanical
fatigue approach [12]. Subsequent fleet inspections revealed that core failure was
observed prior to skin failure and it was assumed that the skin failure was in fact
caused by a breakdown in the sandwich panel construction. The core was replaced
with a higher density variety, with higher shear strength and moduli. This modification
has been in service for several years with no reported failures. The modification
represented only a moderate weight increase of the panel, without recourse to
changing skin thickness, which would have proved very expensive and resulted in a
substantial weight penalty.
As mentioned above, another intake duct, of carbon composite construction, also
began to suffer from core failure. The panels of this duct had a carbon backing skin
while the facing skin had a wire mesh bonded to an open weave carbon sheet. The only
similarity was the use of the same density of honeycomb core (although of different
cell size and depth). For other reasons this core had been replaced by a heavier variety,
prior to the discovery of the core failures and the impact of the failures was minimised.
996
Limited data is available on similar failures and only 2 other cases, regarding nacelle
intake barrels, appear to have been documented [13 & 14], however neither case
involved sandwich panels.
A number of theories had been put forward as to the cause of the failures. These
included neighbouring cells resonating out of phase, cell walls resonating or possibly
the panel vibrating as a 2 degree of freedom system (the facing and backing skins
acting as the masses, with the core as the spring) - this phenomenon had originally
been investigated by Mead [15].
4.0 Physical Testing
A number of tests were carried out with "beam" type high cycle fatigue specimens and
also small segments of intake barrel. None of these tests were able to reproduce the
failures observed in service (Figure 1.0) which further served to reinforce the belief
that the failures were attributed to an acoustic mechanism as opposed to a mechanical
vibration mechanism, however in an attempt to cover all aspects it was decided to
carry out a progressive wave tube (PWT) test on an abbreviated panel.
For simplicity it was decided to test a flat sandwich panel of overall dimensions
36"x21" (Figure 2.0). The panel was instrumented with 12 strain gauges and 2
accelerometers. Two pressure transducers were also mounted in the fixture
surrounding the specimen.
Testing was carried out by the Consultancy Service at the Institute of Sound and
Vibration Research (IS VR) at the University of Southampton.
4.1 PWT Results
The panel was first subject to a sine sweep from 50 to 1000 Hz in order to identify its
resonant frequencies. The response of a strain gauge at the centre of the panel has been
included in Figure 3.0. On completion of the sine sweeps, the linearity tests were
carried out.. As only 8 channels could be accommodated at one time, it had been
decided to arrange the parameters into 5 groups, with each group containing 4 strain
gauges, 2 accelerometers, 1 pressure transducer on the fixture and 1 pressure
transducer in the PWT (this was required by the facility for the feedback loop).
The initial tests were carried out with a power spectral density of the applied loading
constant over the 100 Hz to 500 Hz range, however when using this bandwidth only
155 dB overall, could be achieved. In an attempt to increase the strain levels it was
decided to reduce the bandwidth to 200 Hz. The bandwidth (BW) was subsequently
reduced to 100 Hz and finally 1/3 octave centred on the predominant response
frequency of the panel. When failure occurred a dramatic change in response was
observed. The failure mechanism was that of core failure as shown in Figure 4.0. There
was no indication of facing or backing skin distress.
997
5.0 Comparison With Theoretical Predictions
5.1 Fundamental Frequency
From the strain gauge readings the panel was seen to be vibrating with simply
supported edge conditions. Soovere [7] suggests that "effective" dimensions
(essentially from the start of the pan down) be used to determine the fundamental
frequency which is given by;
x\n
This equation is applicable to simply supported panels with isotropic facing and
backing skins, thus for the purpose of applying the above equation, the actual section
was approximated to a symmetric (isotropic) section. The predicted fundamental
frequency is given below. It was observed however, that if the panel dimensions are
taken relative to mid way between the staggered pitch of the fasteners a significant
improvement was achieved (see "Soovere (2)" in table 1). Alternative frequency
estimations using an FE model and an ESDU data item [16] are summarised in the
following table;
Method
Freq.(Hz)
% Error
Measured
228
-
Soovere
274.3
+20.3
Soovere
(2)
213.3
-6.4
FE
239.04
+4.8
ESDU
193
-15.3
Table 1 - Comparison of calculated frequencies for simply supported sandwich
panel.
Note; the percentage error is based on the actual measured response frequency of the
panel in the PWT.
Given that the excitation bandwidth extended (at least initially) up to 500 Hz, modes
up to 500 Hz were obtained from the FE model. In actual fact 2 FE models were used,
the first was a basic model with 380 elements, however a more detailed model, shown
in Figure 5.0, (with essentially each element split into 4) was used for the results
presented in this paper. The predicted modes from the FE model were as follows;
Mode
No.
Frequency
(Hz)
Mode No. in x
direction (m)
Mode No. in y
direction (n)
Figure
No.
1
239.04
1
1
6
2
334.0
2
1
7
3
430.02
1
2
8
4
1
9
Table 2 - Finite Element Model Predicted Frequencies.
998
5.2 rms Strain
The predicted strains were calculated using Blevins' normal mode method (NMM) [5],
with a joint acceptance of unity for the fundamental mode of vibration, using the
following expression;
s^= Lp.(fJ . ^ (2)
In an attempt to improve the estimated response, the rms strain was calculated for
each mode within the bandwidth of excitation. The Joint accetptance for each mode
was calculated using equation 3 and the calculated strains for each mode were then
factored by the relevant joint acceltance term. The overall strain was then calculated
for all the relevant modes. A comparison with ESDI! [16] has also been included,
however the ESDU method does not provide an indication of shear stress in the core.
Soovere presents a simple expression for the joint acceptance function for a simply
supported panel excited by an (acoustic) progressive wave, for the case where n is
odd;
•2 _ ^ (l-Cos(m7c)Cos(c0ra/c) , .
(1 - (cD^a/ mrcc)^)
Note, when n is even the joint acceptance is zero.
Given that the bandwidth varied for the applied loading, the overall SPLs were
expressed as spectrum levels for the purpose of comparison in the linearity results, the
results (both measured and predicted) have been summarised in table 3, (SGI results
have been plotted in Figure 10.0). The results from the ESDU data item [16] have
been included in table 4 for comparison.
Note - due to recorder channel limitations SGI & SG2 were not connected at the time
of failure and no results were available at the highest sound pressure levels.
OASPL
(dB)
Spectrum
Level SPL
(dB)
Measured
Strains(u£)
SGI SG2
Calculated
(ps) j=l
SGI SG2
Calculated
(Multi Mode)
SGI (us) SG2
130
107
7
7
8.7
9.2
2.5
2.6
140
117
20
19
27.5
29.1
8.0
8.1
150
127
55
60
87.0
92.0
25.4
25.8
155
132
100
100
154.7
163.7
45.1
45.8
157
134
130
130
194.8
206.0
56.8
57.6
163
140
-
-
388.7
411.1
202.5
162.7
164
141
-
-
436.1
461.2
312.0
250.7
Table 3 - Comparison of Measured & Predicted rms Strains
for the Panel Centre, Facing & Backing Skin Gauges.
999
OASPL
(dB)
Spectrum
Level SPL
(dB)
Measured
Strains (pe)
SGI SG2
ESDI! Strains
SGI SG2
130
107
7
7
10.1
29
140
117
20
19
150
127
55
60
101
290
155
132
100
179.5
515.7
157
134
130
■E9
253.6
728.5
163
140
-
-
637
1830
164
141
-
-
1010
2900
Table 4 - Comparison of Measured & ESDI! Predictions of the
rms Strains for the Panel Centre, Facing & Backing Skin Gauges.
There is a considerable difference in the calculated response from using a joint
acceptance of unity for the fundamental mode and that when estimating the joint
acceptance for each mode and calculating the overall response for several modes,
however it was observed that if the average value from both methods is used the
response compares favourably with that measured (-at least for the cases under
consideration). The average value has been included on the linearity plot for SGI,
shown in Figure 10.0). In general, the level of agreement between theory and practice
was considered adequate and it was decided to apply the theory to estimating the shear
stresses in the core (Table 5);
OASPL
(dB)
Spectrum
Level SPL
(dB)
Core Shear
Stress (J=l)
(rms psi)
Core Shear
Stress
(Multi Mode)
(rms psi)
Average
Core Shear
Stress
(rms psi)
Peak Core
Shear
Stress
_ (E£!i _
130
107
0.3
0.09
0.19
0.58
140
117
0.94
0.28
1.84
150
127
2.99
0.88
1.93
5.8
155
5.31
1.57
3.44
10.32
157
134
6.69
1.97
4.33
13.0
163
140
13.35
5.57
9.46
28.38
164
141
14.98
8.59
11.78
35.34
Table 5 - Predicted Core Shear Stress.
6.0 Discussion & Recommendations
The ESDI! method proved very conservative and will thus give a degree of
confidence when used in the early stages of the design process. Blevins Normal Mode
Method was observed to give reasonable accuracy in predicting the highest strains in
the panel and would merit use when designs have been fixed to some degree; at which
stage FE models become available.
For panels whose predominant response is in the fundamental mode it is accepted that
the contribution from shear to overall deformation is very small. The main concern
when designing a honeycomb sandwich panel which is subject to "severe" acoustic
1000
loads has tended to focus on skin strains and to some degree the properties of the core
material have been ignored. The fact that low skin strains are observed has the effect of
giving an impression that there is no cause for concern, however when the properties
of the core material are low or unknown, some caution is required. There is
unfortunately no available S-N data for the type of honeycomb used in the construction
of the panel, however the allowable ultimate strength for the core material is of the
order of 26 psi, so clearly the 163 dB level was sufficient to cause a static failure while
the lower SPLs can be assumed to the have contributed to initiating fatigue damage.
On cutting up the test panel, a large disbond was observed however it did not extend
to the panel edge where cracking had occurred (the mid point of the long edge being
the location of maximum shear for a simply supported panel) and it was the opinion of
the materials department that the failure had not initiated in the disbond.
The SPLs used in the test were not excessively high and were comparable to service
environments (an example of which is given in Table 6). It should be noted that while
the levels in Table 5 are 1/3 octave bandwidths, the actual spectrum is not generally
flat within each band for engine intakes, but is rather made up of tones (Figure 11).
These tones or spectrum levels can thus essentially be the band level and thus some
caution should be exercised when converting intake band levels to spectrum levels
using the traditional approach [17].
1/3 Octave Centre
Sound Pressure
Frequency (Hz)
Level (dB)
100
141
125
133
160
140
200
142
250
140
315
139
Table 6 - Typical Acoustic Service Environment.
Note; Overall levels may reach 160 - 170 dB, however they tend to be influenced by
SPLs at blade passing frequencies, which are much higher than panel fundamental
frequencies.
7.0 Conclusion
It has been shown that although moderate levels of acoustic excitation produce quite
low overall rms strains in the skins of honeycomb sandwich panels, it is still possible,
when using very light weight cores, to generate core shear stresses of a similar order of
magnitude to the allowable ultimate strength of the material.
Acknowledgements
The author acknowledges the support of Short Bros. PLC in the course of preparing
this paper and also the assistance of Mr Neil McWilliam with regard to the FE
modelling.
1001
References
I . 0 Air worthiness Requirements (JAR/FAR) Section 25 .57 1 .d.
2.0 Miles, J.W., "On Stmctural Fatigue Under Random Loading, " Journal of the
Aeronautical Sciences, (1954),Vol.21, p753 - 762.
3.0 Powell, A., "On the Fatigue Failure of Structures due to Vibrations Excited by
Random Pressure Fields,” Journal of the Acoustical Society of America, (1958),
Vol.30, No.l2,pll30- 1135.
4.0 Clarkson, B.L., "Stresses in Skin Panels Subjected to Random Acoustic
Loading," Journal of the Royal Aeronautical Society, (1968), Vol.72,
plOOO- 1010.
5.0 Blevins, R.D., "An Approximate Method for Sonic Fatigue Analysis of
Plates & Shells," Journal of Sound & Vibration, (1989), Vol.129, No.l,
p51-71.
6.0 Holehouse, I., "Sonic Fatigue Design Techniques for Advanced Composite
Aircraft Structures," AFWAL TR 80-3019,(1980).
7.0 Soovere, J., "Random Vibration Analysis of Stiffened Honeycomb Panels with
Beveled Edges," Journal of Aircraft, (1986), Vol.23, No. 6, p537-544.
8.0 Acoustic Fatigue Design Data (Part 1), AGARD-AG- 162-72, (1972).
9.0 ESDU International, London, Series on Vibration & Acoustic Fatigue.
10.0 Richards, E.J., Mead, D.J., "Noise and Acoustic Fatigue in Aeronautics," John
Wiley & Sons, New York, (1968).
II. 0 Clarkson, B.L., "A Review of Sonic Fatigue Technology," NASA CR 4587,
(1995).
12.0 Millar, D., "Analysis of a Honeycomb Sandwich Panel Failure," M.Sc. Thesis,
University of Sheffield, (1995).
13.0 Holehouse, L, "Sonic Fatigue of Aircraft Structures due to Jet Engine Fan
Noise," Journal of Sound & Vibration, (1971), Vol. 17, No. 3, p287-298.
14.0 Soovere, J., "Correlation of Sonic Fatigue Failures in Large Fan Engine Ducts
with Simplified Theory," AGAEUD CPI 13 (Symposium on Acoustic Fatigue),
(1972), pi 1-1 - 11-13.
15.0 Mead, D.J., "Bond Stresses in a Randomly Vibrating Sandwich Plate: Single
Mode Theoiy," Journal of Sound & Vibration, (1964), Vol.l, No. 3,
p258-269.
16.0 ESDU Data Item 86024 (ESDUpac A8624), "Estimation of RMS Strain in
Laminated Face Plates of Simply Supported Sandwich Panels Subjected to
Random Acoustic Loading," Vol. 3 of Vibration & Acoustic Fatigue Series.
17.0 ESDU Data Item 66016, "Bandwidth Correction," Vol. 1 of Vibration &
Acoustic Fatigue Series.
1002
Backing Skin
Figure 2.0 - PWT Test Specimen.
1003
Microstrain (dB)
Figure 3.0 - Response of Strain Gauge SGI During Sine Sweep.
Figure 4.0 - Section Through Failure Region in PWT Panel.
1004
Figure 5.0 - PWT Panel Finite Element Model.
Figure 8.0 - FE Mode 1 (m==l, n=2) Figure 9.0 - FE Mode 2 (m=3, n=l)
1005
Sound Pressure Level (dB)
Figure 10.0 - Linearity Plot for SGI (Measured & NMM Prediction).
Figure 11.0 - Typical Spectral Content of Intake Duct Sound
Pressures with Equivalent 1/3 Octave Levels Superimposed.
1006
Time Domain Dynamic Finite Element Modelling in Acoustic Fatigue Design
Authors:
P. D. Green
Military Aircraft
British Aerospace
Warton
A. Killey
Sowerby Research Centre
British Aerospace
Filton
Summary
Advanced Aircraft are expected to fly in increasingly severe and varied acoustic environments.
Improvements are needed in the methods used to design aircraft against acoustic fatigue. Since fatigue
life depends strongly on the magnitude of the cyclic stress and the mean stress, it is important to be able
to the predict the dynamic stress response of an aircraft to random acoustic loading as accurately as
possible.
The established method of determining fatigue life relies on linear vibration theory and assumes that the
acoustic pressure is fully spatially correlated across the whole structure. The technique becomes
increasingly unsatisfactory when geometric non-linearities start to occur at lugh noise levels and/or
when the structure is significantly curved. Also the excitation is generally not in phase across the whole
structure because of complex aerodynamic effects.
Recent advances in finite element modelling, combined with the general availability of extremely fast
supercomputers, have made it practical to carry out non-linear random vibration response predictions
using time stepping finite element (FE) codes.
Using the time domain Monte Carlo (TDMC) technique it is possible to model multi-modal vibrations
of stiffened aircraft panels without making the simplifying assumptions concerning the linearity of the
response and the characteristics of the noise excitation.
The technique has been developed initially using a simple flat plate model. This paper presents some of
the results obtained during the course of this work. Also described are the results of a study of the
“snap-through” behaviour of the flat plate, using time domain finite element analysis. For simplicity, it
was assumed that the dynamic loading was fully in phase across the plate.
Introduction
Aircraft structures basically consist of thin, generally curved, plates attached to a supporting framework.
During flight these stiffen^ panels are subjected to a combination of static and dynamic aerodynamic
loads. On some aircraft there may be additional quasi-static thermal loads due to the impingement of jet
effluxes in some areas. Parts of advanced short take of and landing (ASTOVL) .aircraft may be required
to withstand noise levels up to 175dB and temperatures up to 200deg C. Under these conditions the
established methods of dynamic stress analysis for acoustic fatigue design are inappropriate and cannot
be employed.
British Aerospace (BAe), Sowerby Research Centre (SRC) and Military Aircraft (MA) have been
developing a method to predict the stress/strain response of aircraft structures in these extreme loading
situations. The primary .consideration has been the requirement to create an acoustic fatigue design tool
for dealing with combined static and dynamic loads, including thermally generated “quasi-static” loads.
1007
The resonant response of thin aircraft structures to aeroacoustic loading is generally in a firequency
range which implies that, if defects form, they will quickly grow. Hence to be conservative, it is
generally assumed that a component has reached its life when it is possible to find quite small defects
by non-destructive evaluation techniques. Several different materials and construction methods are used
in modern aircraft and so there are a number of possible failure criteria. In the case of metals, it is the
presence of cracks larger than a certain size. For composites it can be the occurrence of either cracking
or delamination. Degradation due to the presence of microcracks may be monitored by measuring the
level of stiffness reduction which has taken place.
This philosophy simplifies the type of stress analysis needed, because it is not necessary to model
structures with defects present. Materials can be assumed to have simple elastic properties which
remain unchanged throughout their lives. In consequence, it is necessary to know the fatigue behaviour
in terms of a direct relationship between number of cycles to failure and the magnitude of the “nominal”
cyclic stress, or strain, at a reference location.
If considered important and capable of satisfactory treatment, the relationship can be modified to take
into account material property changes due to the development of very small defects at points of stress
concentration. For example, metal plasticity in the region of a small crack, could be included in an
analysis of the stress distribution around a fastener hole. It is well known that plasticity reduces the
peaks of stress which are predicted at defects by analysis which assumes perfectly elastic material
behaviour.
The technique developed at BAe for modelling high acoustic loads combined with possible thermal
buckling uses the time domain Monte Carlo (TDMC) technique together with finite element analysis by
proprietary FE codes. Response characteristics are predicted directly in the time domain using
simulated random acoustic loadings. These may then be used in fatigue life estimations which employ
cycle counting methods such as Rainflow counting. It is now practical to predict the vibrational
response of stiffened aircraft panels without the necessity to assume a linear response, and without
simplifying the spatial and temporal representation of the noise excitation.
Since the technique uses proprietary finite element codes, quite large and complex models of aircraft
structure can be analysed in a single run. Standard pre- and post-processor techniques are available to
speed up generation of the finite element mesh and to display the stress/strain results.
The initial development work was carried out by modelling the random vibration of a flat plate. For
fully in-phase random loading at low noise levels the predicted response is predominantly single mode
and at fhe frequency calculated by linear theory. However, as the decibel level is increased, the
frequency of the fundamental rises due to geometrically non-linear stiffening. At veiy high dB levels the
predicted response becomes multi-modal; the resonance peaks move to higher and higher frequencies
and broaden.
The effect of static loading on the response has been studied as part of these investigations to assist in
the validation of the methodology being developed.
Thermal Effects
In some flight conditions it is possible for a panel to be buckled due to constrained thermal expansion
and also be subjected to very noise levels at the same time. An example is when a ASTOVL aircraft
hovers close to the ground for an extended period, panels which are initially curved, or thermally
buckled panels may possibly be snapped through from one side to the other by a large increase in
dynamic pressure.
“Snap-through” can be potentially damaging to the structure of an aircraft if it occurs persistently,
because the process is associated with a large change in the cyclic bending stress present at the edge of
a stiffened panel. High performance aircraft must therefore be designed so that snap-through never
occurs in practice.
1008
The dynamic response of curved panels or buckled flat plates is difficult to predict theoretically because
of non-linear effects. The established acoustic fatigue design techniques, which are based on linear
vibration theory, are only able to provide approximate predictions of the loading regimes in which
particular panels might be expected to undergo snap-through.
The TDMC method can be used to model non-linear multi-modal vibrations of stiffened aircraft panels
which are also subject to quasi-static stress. In particular calculations may be carried out in the post-
buckling regime.
With this technique simulated random dynamic pressure loading, with measured or otherwise known
spectral characteristics is applied to a curved, or post-buckled panel and the time domain response
calculated. The magnitude of the dynamic loading may then be increased until persistent snap-through
is observed in the predicted response. This gives the designer the ability to design out the potential
problem by systematically altering the most important parameters in order to identify the critical
regime.
Fatigue Life Estimation
Although acoustic fatigue is a complex phenomenon, it has been established that the life of a component
mainly depends on its stress/strain history. The most important factors in this regard are the magnitude
and frequency of the cyclic strain and the mean level of stress at the likely failure points. On this basis
fatigue life can be estimated by carrying out the three stage operation illustrated in Figure 1.
Stage 1: Determine Loads
A determination of static design loads is relatively straightforward compared to a calculation of th.e full
temporal and spatial dependence of the aeroacoustic pressure on a military jet in flight. This is an
enormous task in computational fluid dynamics (CFD). Designers have to rely on experimental data
which can come from measurements on existing aircraft or from scale model tests of jets, for example.
Existing databases can be extrapolated if the circumstances are similar. Experimental noise data is
usually in the form of power spectral density curves as opposed to time series fluctuating pressures, but
either can be used, depending on the circumstances.
Stage 2: Calculate Stresses
The technique chosen to obtain the stresses clearly depends on how much knowledge there is about the
expected loads. In the early stages of design analytic techniques would be used to establish approximate
sizes and stress levels. However, later on when the design is nearly completion, finite element (FE)
stress analysis can be used to model the effect of random acoustic loading on the parts of the skin which
are likely to be severely affected. These calculations would, of course, be done including the effect of
attached substructure.
The established method of designing against acoustic fatigue uses a frequency domain technique which
relies on the validity of linear vibration theory. The method forms the basis of a number of methodology
documents published by the Engineering Sciences Data Unit (ESDU). Whenever there^^e large out-of-
plane deflections the frequency domain method cannot be used because of the “geometric non¬
linearity”. From a strictly theoretical point of view such analyses have to be carried out in the time
domain, although approximate methods are applied with some success.
The established technique produces inaccurate results for curved panels, buckled panels and for panels
under high amplitude vibration. Geometric non-linearity usually stiffens a structure in bending so there
is a tendency to overestimate the stress levels using the frequency domain technique. This conservatism
is clearly useful from the point of view of safety, but it can lead to possible “over-design”.
Unfortunately this is not always the case when there are compressive static stresses present. The
established method also fails if the phase of the noise varies significantly over the surface of the
structure, which is the case in a number of aeroacoustic problems. The techniques under development
are designed to overcome these problems.
1009
Stage 3: Estimate Fatigue Life
In cases of random acoustic loading it is customary to assume that damage accumulates according to the
linear Miner's rule. Fatigue life is determined from experimental data in the form of stress (or strain
amplitude), S, versus number of cycles to failure, N. If a number of cycles, n, of stress/strain, S, occur at
a level of stress/strain where N(S) cycles would cause failure then the fractional damage done by the
n(S) cycles is n(S)/N(S).
Various methods have been developed for obtaining n(S) from the stress (strain) response. If the
excitation is stationary, ergodic and the response is narrow band random then the function n(S) can be
shown to be in the form of a Rayleigh distribution and the damage sum can be evaluated from plots of
root mean square stress (or strain) against number of cycles to failure. If the statistics of the response
are not Gaussian then it is necessary to count the numbers of stress cycles from the time domain
response and use constant amplitude S/N curves. It is now widely accepted that the best way of
counting the cycles is to use the Rainflow method, [1].
The Loading Regime
The loads on an aircraft may be conveniently divided into static and dynamic.
Loads which vary only slowly are:
a) Steady Aerodynamic Pressure Loading,
b) In-Plane Loads transferred from “external structure”, and
c) Thermal Loads due to Constrained Expansions.
The rapidly varying loads are, of course, the aeroacoustic pressure fluctuations which originate from
any form of unstable gas or air flow.
This division is central to the methodology which has been developed because it enables the modelling
to be carried out in two distinct phases. The, so called, static loads do vary, of course, but the idea is to
separate effects which occur on a time scale of seconds from the more rapidly varying acoustic
phenomena. The aim is to split the loads so that the quasi-static effects can be calculated in an initial
static analysis which does not depend on a particular dynamic loading regime. Any aerodynamic
pressure may be divided into a steady part and a fluctuating part. The natural place to make the cut-off
is at IHz which means that epoch times for TDMC simulations are then of the order of a second. The
epoch time must not be too short because of statistical errors, and it cannot be too long because this
would invalidate the assumption of constant quasi-static loads. In practice, there is another constraint on
the epoch time. The number of finite elements in the model coupled with the premium on cpu time
places an obvious limit on the epoch time.
Comparison of the Time and Frequency Domain Methods
A flow chart comparing the two methods is given as Figure 2. The main difference between the two
techniques lies in the representation of the dynamic loads. The FD method uses rms loadings and
spectral characteristics, whereas the TD method uses the full time series loadings. Gaussian statistics
are, de facto, assumed by the FD method, but this is not necessarily the case with the TD technique.
Application of the frequency domain method requires that the response is dominated either by a single
mode or a small number of modes. To determine whether or not this is the case in practice, a normal
modes analysis must be followed by a determination of the amount of coupling between the excitation
and each mode. This can be determined quite accurately even if there is a certain amount of potential
non-linearity by computing the joint acceptances for each mode, which are overlap integrals of the
mode shape functions with the spatial characteristics of the excitation. Normally these quantities will be
dominated by a few of the low order modes. If there is significant coupling into more than one mode
then it will be necessary to use the TD method instead.
With the time domain technique it is possible to represent the dynamic loads in a way which models the
convection of the noise field across the structure. Very complicated loadings can be applied to large
1010
models but in consequence it can be difficult to validate the results obtained, because they cannot be
checked against anything other than test data which is itself subject to confidence levels. In addition it
must be remembered that the TDMC results themselves are subject to statistical variability. Finally it
should be noted that TDMC data must be used in conjunction with constant amplitude endurance data.
Rms fatigue data can only be used with frequency domain results.
Time Domain Finite Element Modelling
Until recently, the majority of finite element analyses were applied to static loading conditions or “low
frequency normal modes analysis”. The method involves the use of an implicit code to invert in one
operation, a single stiffness matrix, which can be very large. The general availability of extremely fast
super-computers has now made it possible to carry out large scale non-linear dynamic finite element
modelling using explicit FE codes. These codes use very similar types of element formulation to the
implicit ones, e.g. shells, solids and bars, but the solution is advanced in time using a central difference
scheme.
One potentially very useful capability of time domain modelling is the application of acoustic pressure
loadings which vary both in the time and spatial domains. If the spectral characteristics are known,
either from test or from other modelling it is possible to generate samples of random acoustic noise and
apply these directly to the finite element model as a series of “load curves”.
The technique for determining time series noise was developed by Rice [2] and Shinozuka [3]. They
showed that homogeneous Gaussian random noise can be generated from the power spectral density as
a sum of cosine functions with different frequencies and random phase. Noise can be temporally and
spatially correlated noise by deriving phase differences from cross spectral functions if they are known.
The TDMC method can be quite costly in terms of central processor unit (cpu) time because the
solution must be recalculated at each point in time. To reduce execution times, the explicit codes
employ reduced numbers of volume integration points in the finite element formulations. However in
this work cpu times are extended because long epoch times are required to ensure adequate statistics. It
can take more than 24hrs to obtain a solution over a half second epoch if there are a few thousand
elements in the model.
Hence there is always a practical limit to the size of a particular time domain finite element analysis,
(TDFEA). If the loading and geometry are not too complicated, the frequency domain method of
analysis can be tried initially to gain more understanding of the nature of the response in an
approximate way. In some cases the vibrational response regime must be considered carefully to decide
whether TDFEA is really necessary. These may be situations where the non-linear effects are only
moderate.
It would be ideal if the full dynamic response of an aircraft could be determined with a fine mesh model
in one huge operation, but experience has shown that this requires too many elements. It is possible to
construct frill models with reduced stiffness using superelements, enabling flutter and buffet to be
studied, because these are essentially low frequency phenomena. However, in time domain analysis it
has been found that models containing a large amount of detail, such as fasteners and individual
composite material plys, require a great deal of cpu time. To progress we must devise some strategies to
overcome this situation. Since a full TDFEA can only handle a part of the aircraft structure, it is very
important that loads external to the area under consideration are properly taken into account. This is
cmcial to the success of this type of modelling as it is to all finite element modelling.
The most important parameter in any time series analysis is the time step. This is determined by the
velocity of sound waves in the structural material, and is generally of the order of a/v where a is the
shortest element dimension and v is the velocity of longitudinal sound. A small time step is therefore
required when the elements are small and the velocity of sound is large. For an aluminium model with
10mm square elements the time step is about 1.6ixs. Hence a TDMC run with a half second epoch time
needs about a half a million steps. A simple 5000 shell element calculation on a Cray C94 would take
approximately 10 hours.
1011
Dynamic FE models of aircraft structure can be constructed in many ways, using shell elements, beam
elements and/or solid elements. Special elements exist for damping and for sliding interfaces. Joints can
be modelled with sliding interfaces, or with short beams, or just with tied nodes. Fasteners can be
modelled with small solid elements, with short beams or with tied nodes, also. Unfortunately, however
short beams and small solid elements cause a dramatic lowering of thC time step. For example, if the
smallest fastener dimension is, say 3mm, the time step will have to be reduced to about O.Sfis if solid
elements are used in the model. The effect on cpu time is such as to make the calculations impractical.
Sliding interfaces are an efficient way to model skin/substructure contact in explicit analyses, but it is
important to choose the algorithm carefully because some techniques can consume large amounts of cpu
time.
The best practical way of representing stiffened aircraft panels for TDMC analyses is considered to be
with four noded shell elements simply tied together at their edges. A number of efficient shell
formulations are avail-able and meshes can be rapidly produced from the design geometry. Of course,
such models cannot be expected to produce highly accurate stress data in the region of small features
but this aspect has to be sacrificed in the interests of achieving statistically significant amounts of time
series data. To improve the accuracy of stress predictions in the neighbourhood of stiffeners etc., it will
be necessary to couple TDMC analyses with fine mesh static analyses.
The Generation of Time Series Data
A number of factors must be borne in mind when generating time series data for TDMC calculations. It
is important to consider carefully the frequency range and number of points which define the load
spectrum in conjunction with the epoch time and number of points on the time series.
The Nyquist Criterion [4] states that the time increment must be less than or equal to one over twice the
upper frequency on the power spectral density curve. For the sake of argument, take the upper
frequency to be IkHz. This means that the time increment must be less than 500ps. A more
conservative time increment is based on the requirement to represent the dynamic response of the
structure as accurately as possible over a full cycle. Assuming a resonant frequency of 500Hz, which is
perhaps near the limit in practice, and 10 points per cycle which is more than sufficient, the lower limit
on the time step works out at about 2C)0|is. Taking all these factors into consideration, the number of
points on the spectrum curve should be of the order of 1000 and there should be between 1000 and
5000 on the time series. Longer epoch times can be used but for reasons of practicality and statistics it
is better to run more than one short epoch simulation rather than one long simulation.
Explicit FE modelling frequently requires that the time step be smaller than 200jj,s. In the example
given above the time step required by the explicit code was 1.6fis. Under these circumstances the
random noise could be defined with a smaller time increment, but going to this level of effort has been
found to produce no measurable change to the calculated response.
Static Initialisation
There are two possible ways of dealing with the effect of static loads in TDMC modelling. Firstly the
complete analysis can be carried out using the explicit code. To do this it is necessary to apply only the
static loads to the model and run the code until equilibrium is reached. By introducing a high level of
artificial damping the stresses created can be relaxed in a relatively short period of time. The time
required depends on the lowest resonant frequency of the structure and the size of the smallest element
in the model. This facility is termed “dynamic relaxation”.
The alternative is to make use of another facility in the explicit code called “static initialisation”. The
deformed shape and stress state of the structure with just the static loads applied are first obtained very
quickly using an implicit code. The solution for the stressed state is then initialised into the explicit
code prior to the application of the dynamic loads. Dynamic relaxation may be used to smooth out any
differences between the models.
1012
Damping Representation
Vibrating aircraft structures are damped by several mechanisms, for example friction at joints, re¬
radiation of acoustic waves, and energy loss in viscoelastic materials It is difficult to generalise about
the relative importance of each damping process in practice. Also reliable quantitative data is not
available in sufficient detail to justify the inclusion of complex models of damping into the TDMC
analyses. Test results on vibrating stiffened aluminium panels tend to show that the damping is best
approximated by a combination of mass and stiffness proportional coefficients. There is a range of
frequencies in which the damping ratio can be considered to be roughly constant. Until more detailed
experimental data are available the most expedient approach is to assume a nominal value for the global
damping ratio which does not change with frequency. Over the years it has become standard practice to
assume a damping ratio of about 2% for fastened aluminium structures.
Equivalent Linearisation
There are some loading regimes in which the non-linear response to high levels of random acoustic
loading can be approximately found using a linearisation technique combined with a frequency domain
analysis. The basic idea is to replace the non-linear stiffness term in the general vibration equation by a
linear term such that the difference between the rms response of the two equations is minimised with
respect to a shifted fundamental resonance frequency. If an approximate equation for the non-linear
stiffness is known then it is possible to derive an expression for the shifted “non-linear” resonance
frequency. The rms response to random acoustic loading may then be found by combining the
Miles/Clarkson equation with some form of static geometrically non-linear analysis. References to this
technique are Blevins [5], Mei [6] and Roberts & Spanos [7]. Where the geometry is complex the most
appropriate form of analysis is clearly finite element analysis.
Implementation and Validation Studies
The stress analysis work described in this paper has been undertaken using MSC-NASTRAN and
LLNL-DYNA.
NASTRAN is a well known implicit finite element code which is capable of handling very large
numbers of elements. It has been developed very much with aerospace structural analysis in mind. It is
basically a linear analysis code, although there are a large number of adaptations to deal with non-linear
problems. It can also function as a dynamic code, but is much slower than DYNA in this mode because
it basically needs to solve the complete problem at each time step. The non-linear features which are
most relevant to the type of stress analysis being discussed here are those concerned with geometric
non-linearity. Geometric non-linearity is treated by dividing the load into a series of steps, obtaining the
solution incrementally. In this work, the code has been used for linear and non-linear static analyses and
for normal modes analyses.
DYNA is an explicit finite element code originally developed for the calculation of the non-linear
transient response of three dimensional structures. The code has shell, beam and solid element models
and there are a large number of non-linear and/or anisotropic material models available. DYTMA was
developed primarily for the modelling of impact and there is no limit, as far as the code is concerned,
on the size of finite element model which can be analysed. Many of its advanced features relate to
impact modelling and are not required for this work. One useful feature, however is the laminated
composite material model based on the equivalent single layer approximation. This code has been used
for the TDMC calculations presented in this paper.
Finite element models for the stress analyses were produced using MSC-PATRAN, It has a wide range
of geometry and mesh generation tools and now has built in interfaces for both NASTRAN and DYNA.
The element definitions are compatible with both codes and it is a simple matter to toggle between the
two codes by changing the analysis preference. Not all the features of DYNA are supported and some
of the parameters must be set by editing the DYNA bulk data produced by PATRAN.
NASTRAN results were post-processed using PATRAN. DYNA results were post processed using
TAURUS, which is faster and easier to use than PATRAN for this task; Some special in-house codes
have been written to generate random acoustic noise from power spectral densities, as described above,
1013
and to post process time series output from TAURUS. One of the codes incorporates a fast Fourier
transform (FTT) routine to determine spectral responses from the DYNA time series predictions. These
codes are covered under the generic title “NEW-DYNAMIC”.
TDMC Calculations on a Simple Flat Plate
Calculations have been performed on a very simple model to implement the TDMC technique and
develop the in-house software referred to above. A PATRAN database was constructed representing a
simple flat plate, 350mm x 280mm x 1.2mm thick as an array of shell elements 34x28. For simplicity
the boundary conditions were taken either as simply supported or clamped. There are a number of
alternative shell element formulations available in DYNA, [8]. The Hughes-Liu shell was used initially
because of its good reputation for accuracy, but later a switch was made to a similar, but slightly faster
shell element, called the YASE. It was found that equally satisfactory results could be obtained more
quickly using this element.
Analyses without Static Loads
A series of DYNA calculations were carried out with a fiiUy correlated random acoustic pressure load
with a flat noise spectrum between OHz and 1024Hz. Investigations were carried out into the effect of
varying the sound pressure level, the epoch time, the mesh resolution, the damping coefficient and the
stochastic function.
Figure 3 shows the displacement response of the central node of the model for a sound pressure level of
115dB (about 12Pa rms), simply supported edges and mass proportional damping set so that the
damping ratio was equal to 2% at the fundamental (1,1) resonance of the plate. The corresponding
spectral response is shown in Figure 4. shows a sharp resonance peak at a frequency of 61.0Hz which is
very close to the theoretical frequency of the (1,1) mode for the simply supported plate. The in-phase
loading means that only the modes with odd numbered indices are excited. I^e peaks corresponding to
the (3,1) and (1,3) modes are, however, not visible on the plot because they are too small. It may be
concluded from these results that the behaviour of the plate at these pressure level is well within the
linear regime.
An investigation into the behaviour of the rms displacement response as a fijnction of SPL was carried
out by increasing the loading incrementally from 75dB (0.1 2Pa rms) to 175dB (12kPa rms). The results
are shown in Figure 5. Also shown are theoretical predictions obtained using the Miles/Clarkson
formula with NASTRAN linear and non-linear analyses as explained above, see below for discussion.
The statistical variation of the results was investigated by repeating a half second epoch TDMC run ten
times with different samples of flat spectrum noise. It was found that the standard error of the rms
response was about 16%. A second set of ten repeats were carried out with the epoch increased to 2.5s.
In this case the standard error reduced to roughly 8%. From the theory of stochastic processes, it can be
shown that the standard error is inversely proportional to the square root of the epoch time. On this
basis therefore the ratio between the standard errors should be equal to the square root of five, or 2.23.
From the analyses this ratio is about 2. Further runs established that these results are not affected by the
vibration amplitude, even when the response becomes non-linear.
Cautiously therefore, it can be concluded that the variance of the TDMC results is unaffected by non¬
linearity of the response. This is an important finding because it builds confidence in the technique. In
many practical situations it may be necessary to rely on just one simulation and an appropriate factor of
safety. It can be quite time consuming to carry out a large number of repeat TDMC simulations. The
level of variance would be first established by repeating one load case a number of times, before
confidently applying it to the results of other load cases.
Comparisons with Linearised Theory
The linear theory of plate bending, [9], leads to relationships between the central deflection, w, of a
rectangular plate and a uniform static pressure load, Pstat which take the following form.
Psutab = kcffW (1)
1014
where a and b are the length and breadth of the plate, and k^ff is an effective stiffness parameter which is
a function of the modulus of rigidity of the plate and the edge boundary conditions. For the plate
studied k^ff is about 30N/mm for the case of simply supported edges and lOON/mm for clamped edges.
The above equation only holds, however, at very low amplitudes, as can be seen from Figure 6. This
compares geometrically non-linear NASTRAN predictions with the linear ^eory. Curves are shown for
both simply supported and clamped boundary conditions. The finite element results show the
characteristic hardening spring type of non-linearity.
At higher amplitudes the dynamic behaviour may be approximately predicted using “equivalent
linearisation” theory, which assumes that the response remains predominantly single mode, but with a
resonant frequency which rises as the stiffness of the structure increases. When the deflection is large
the static force-deflection relationship can be written as the sum of a linear stiffness term and a cubic
non-linear term:
psiatab = kw(l+pw^) (2)
where a b is the force, k is the linear stiffness. The equation is written with the leading term factored
out to emphasis the point that p. is a constant which is small compared to the rms deflection. In the limit
of small w we can expect the pw^ term in the brackets to be negligible compared to one, which means
that the k in this equation must be the same as kcff above.
Equation 2 was fitted to the NASTRAN results shown in Figure 6 to find the best fit values of k and p.
Table 1 shows the results compared with the effective stiffness calculated from linear plate bending
theory. It can be seen that the theoretical stiffness is almost identical to the best fit k from the non-linear
finite element analysis.
With reference to the results in Figure 5, it is obvious that the nature of the response is strongly
dependent on the amplitude of the vibrations. For rms displacements up to about 4% of the plate
thickness the behaviour was completely linear. For displacements between 4% and 150% of plate
thickness, the response was essentially single mode dominated but the level could not be predicted by
the Miles/Clarkson approach. The “equivalently linear” solution does, however, agree with the DYNA
result up to a displacement of about l.8mm. The linearisation approach cannot be expected to be
correct for displacements above about 1.5 times plate thickness. Above this point the response predicted
by DYNA was multi-modal and strongly non-linear. The equivalently linear predictions departed
considerably from the DYNA results when the vibration amplitude was very high.
It was also observed that the frequency response peaks became increasingly noisy for higher pressures,
representing the increased level of non-linearity in the plate vibrations. The increase in the frequency of
the fundamental mode with acoustic pressure, as calculated by DYNA, is shown in Figure 7. Predictions
from equivalent linearisation theory and from the theory of Duffing's equation are also included, see
Nayfeh & Mook [10].
t„, = f(l + 3nw™,Y’
Equiv. Lin. Pred.
(3)
f„, = f(l+(3/8)w„,^)
Duffing's Eq. Pred.
(4)
The DYNA results lie mostly between the two theoretical curves, agreeing particularly well with the
results of equivalent linearisation theory up to around 700Pa (151dB rms). The level of agreernent
obtained shows that the frequency response behaviour of the DYNA model is similar the theoretical
predictions, providing an independent check on the results. As might be expected, at around 700Pa the
agreement begins to breakdown, since the linearisation theories are not valid for deflections which are
significantly greater than the plate thickness. It may be concluded, however, that the effect of geometric
non-linearity at high amplitudes is being computed by DYNA in a reasonably accurate manner. A
detailed comparison with experimental data is needed to determine the accuracy of the DYNA response
predictions themselves.
1015
Due to the increasingly irregular shape of the frequency response functions derived from the DYNA
time series predictions at high acoustic loads, it was not possible to calculate very accurate peak widths
for pressures above approximately 135dB (I20Pa rmsX Figure 8 shows that the width of the peak
increased with increasing acoustic pressure, but not in a regular manner. When the damping is mass
proportional, equivalently linear theory predicts that the width of the peak should remain unchanged as
the pressure rises. This is because the geometric stiffening effect of rising acoustic pressure exactly
cancels the effect of a smaller damping ratio at the higher resonant frequency. This graph shows this as
a horizontal straight line at 2.44Hz. The DYNA result is closer to the type of behaviour observed
experimentally where the width of the peak generally increases with increasing the sound pressure level.
Analyses with Combined Loads
Further work was conducted with static loads superimposed on different levels of random acoustic
loading. These calculations were done using the coupled NASTRAN-DYNA approach outlined earlier.
That is to say the deformed geometry was obtained by applying the static loads to a NASTRAN model,
with the results being initialised into DYNA and dynamically relaxed before the dynamic loading was
applied. Calculations were performed with compressive in-plane loading, static pressure loading and
thermal loading. With the exception of the thermal runs, the boundary conditions used in these runs
were identical to clamped, except that symmetrical in-plane movement of the edges was permitted. We
have called these conditions “semi-clamped”. It has been found that the fundamental resonant frequency
of the plate without static loading is only reduced by a very small amount if the appropriate in-plane
degrees of freedom are released, see Figure 9. These boundary conditions are actually closer to those
which exist in reality when a panel in built into a larger structure.
Figures 10-14 show results of some of the analyses which have been carried out. They give time series
data along with spectra responses calculated by the in-house post-processing code. Numerical data
derived from these results are summarised in Tables 3-5.
A series of analyses have been carried out with compressive in-plane loads equal to one third of the
theoretical buckling loads in compression. For the plate used, the forces per unit side length were -
3.46N/mm in the x-direction and -5.46N/mm in the y-direction. The results of one analysis are shown in
Figure 10. It has been found that the response remains dominated by the fundamental (1,1) mode as
long as the plate is unbuckled and the SPL is low. The softening effect of the compressive loads on the
frequency agreed quite well with Rayleigh-Ritz predictions, [5], up to an SPL of ?dB. At higher sound
pressure levels, the DYNA results reflected stiffness changes which were greater than those predicted
by the theory. The- same was found in the case of tensile loading. It is believed that these differences are
due to approximations built into both the Rayleigh-Ritz theory and the DYNA code.
Figure 1 1 shows the results of a calculation with a superimposed normal pressure. The magnitude of the
pressure, 700Pa, was chosen so as to provide an example of “post-buckled” analysis. This size of
pressure causes the plate to bow out in the centre by about 0.6mm. It is well known that in the post-
buckling regime the random response of a plate depends upon the magnitudes of both the static and
dynamic loads. In this case the static loading was large compared to the applied dynamic loads and
“snap-through” did not occur. The plate simply vibrated about its statically deflected position in the
fundamental mode with a slightly increased frequency.
To provide a test of the DYNA thermal stressing capability, and to carry out an investigation into
“snap-through”, several analyses were carried out with a uniform temperature rise of lOdeg C applied to
the plate with clamped edges. This is quite sufficient to cause buckling because the resulting
compressive biaxial stress, c, is well above the buckling level, Gb- If f is the frequency of the
fundamental and J is a constant equal to 1.248 because of the clamped boundary condition, the two
stresses can be determined approximately from
a = EaT/(l-v^)
(5)
Cb = 4pa^f2/J
(6)
1016
where E, a, v and p are Young's modulus, coefficient of thermal expansion, Poisson's ratio, and density
respectively. Using these formulae we find c = 24MPa and Ob = 14MPa.
Analyses were carried out with several different levels of dynamic load. The results of three of the
calculations are shown in Figures 12,13 and 14. It was found that the threshold for snap-through
occurred at an acoustic load of about IkPa, see Table 4. Below this level the mean deflection, w, is a
function of the static load alone, equal to about 2.8nun (the negative sign indicates that the plate has
bowed in direction of negative z). At higher SPLs the mean deflection reduces because the plate snaps
backwards and forwards between positive and negative z. The calculated response spectra for these
higher level runs, show an additional peak at a very low frequency, ie less than lOHz. This is an artifact
caused by the snap-through since the fundamental resonance of the clamped plate is at 1 13Hz.
Figures 10-14 all show probability density functions derived from the time series data. The fluctuations
on these plots are caused by the smallness of the epoch time. In all cases, except for the thermal
calculations with the two largest acoustic loads, it can be seen that the functions are basically Gaussian
in shape. It may therefore be concluded that it is reasonable to assume that the response of a plate in the
post-buckled region is Gaussian unless there is a large amount of snap-through.
Discussion
The work described is the starting point for investigations and validations using more complex FE
models. Further work has been carried out using models including curvature, sub-structure and detailed
features. It is difficult to validate the predictions obtained from such models by comparing with test data
because the results themselves are open to interpretation. It has been found that the predicted stress
levels are closer to the test results when the chosen location is away from any small features. The lack
of good agreement in the neighbourhood of the features can be explained by the relatively coarse mesh
used in the dynamic models. The overall level agreement was much better than that between predictions
based on linear or equivalently linear theory and test. On the basis of experience, the latter tend to over¬
predict by upwards of factors of two and three. From this work it has been found that the DYNA
predictions tend to be greater than test by amounts which vary but are generally much less. The average
over-prediction was about 40% with a significant change as a function of location.
TDMC runs can take a significant amount of computer time to carry out and it is believed that to make
further improvements the technique should be combined with detailed stressing using static finite
element analysis. Inaccurate results can be obtained if the boundaries of the part of the structure under
analysis are not properly restrained. In the case of models of aircraft panels this may significantly affect
the resonant frequencies which in turn affects the level of calculated dynamic stress. In-plane loads on a
panel, perhaps due to thermal stressing, can alter the fundamental by as much as 100-200Hz. Looked at
from a theoretical point of view, the only way to solve this problem is to construct a second, coarse
model of the component, along with some of its surrounding structure. An initial calculation can then be
carried out with this model in order to obtain the loads and boundary conditions for subsequent
application to the original model.
The dynamic phenomenon of “snap-through” cannot be modelled using existing methods and so the
TDMC / finite element technique offers the engineer a way to determine where the likely regions of
unstable vibration are located in circumstances where the structure is complicated by attachments etc.
Conclusions
This paper has sought to explain how time domain finite element modelling can be used to assist in the
design of aircraft against acoustic fatigue. Although the technique is computationally intensive, it does
have a place in the effort to understand complex vibrations, such as the response of structures to
spatially correlated jet noise excitations, or interactions between high sound pressure levels and thermal
loads.
The work at BAe is continuing in an attempt to provide the analyst with a greater ability to determine
dynamic stress levels in advanced structures with complex loadings.
1017
References
1. Dowling N. E. , Fatigue Prediction for Complicated Stress Strain Histories, J Materials 1, 71
(1972).
2. Rice, In Selected Papers on Noise and Stochastic Processes, Ed N Wax pplSO, Dover New York
(1954).
3. Shinozuka M. , Computers and Structures, 2, 855, (1972).
4. Bendat J. S. and Piersol A.G. , Engineering Applications of Correlation and Spectral Analysis
Wiley (1990).
5. Blevins R.D, , An approximate method for sonic fatigue analysis of plates and shells, J Sound and
Vibration, 129, 1, 51 (1989).
6. Mei C. and Paul D.B. , Non Linear multi-modal response of a clamped rectangular plates to
Acoustic Loading, AIAA Journal, 24, 634, (1986).
7. Roberts J. B. & Spanos P.D , Random Vibration and Statistical Linearisation, Wiley, (1990).
8. Whirley R.G. and Engelmann B.E. , DYNA3D: A Nonlinear, Explicit, Three Dimensional Finite
Element Code for Solid and Structural Mechanics — User Manual, Lawrence Livermore National
Laboratory, UCRL-MA- 107254 Rev. 1, (Nov 1993).
9. Szilard R. , Theory and Analysis of Plates, Prentice Hall, New Jersey.
10. Nayfeh A.H. & Mook D.T. , Non Linear Oscillations, Wiley (1979).
1018
Parameter
lUQgnnggll
HISSH
Linear Theory
NASTRAN
NASTRAN
Simply Supported
30.0
30.7
1.09
Clamped
104.
101.
0.266
Table 1: Values of Parameters fitted to NASTRAN results compared with the
linear theory values.
Static Loads
Acoustic Load
Rrms
(Pa/dB)
Theory ]
1 DYNA
(N.mm
N,
(N.mm
Freq
(Hu)
Wfim
(mm)
■SI
Wrms
(mm)
None
1.2 (95.6)
jnggi
0.00290
114
0.01.2
-3.46
1.2 (95.6)
0.00316
94.1
0.00313
-3.46
643.5 (150.1)
1.69
115
1.08
-3.46
1.2 (95.6)
68.33
0.00359
68.1
0.00368
-3.46
700 (150.9)
68.33
2.09
103
1.42
Table 2: Summary of results of calculations with random acoustic loading
superimposed on compressive in-plane loads.
Static Load
p(Pa)
Acoustic Load
(Pa/dB)
DYNA
None
1.2 (95.6)
114
0.0000
0.01.2
700
None
—
-0.646
—
700
12 (115.6)
115
-0.655
0.0189
700
700 (150.9)
134
-0.477
1.28
5 k
None
—
-2.89
—
5 k
12 (115.6)
179
-2.93
0.0152
5 k
700 (150.9)
173
-2.70
0.999
Table 3: Summary of results of calculations with random acoustic loading
superimposed on static pressure loads.
Temperature
T (<»C)
Acoustic Load
(Pa / dB)
1 DYNA 1
None
1.2 (95.6)
114
0.00256
10
None
—
-2.85
—
10
1.2 (95.6)
234
-2.86
0.0140
10
700 (150.9)
219
-2.80
0.197
10
Ik (154.0)
9.01
2.47
1.02
10
1.2k (155.6)
5.01
-0.717
2.24
10
1.5k (157.5)
3.00
-0,376
2.20
10
2k (160,0)
92.2
0.0185
1.95
10
4k (166.0)
195
0.0568
2.08
Table 4: Summary of results of calculations with random acoustic loading
superimposed on a thermal load.
1019
Figure 2: Flowchart Illustrating the Frequency and Time Domain Techniques
(uiUi),ueujeoB,ds,a ^ O
Displacement of central node for 12Pa rms acoustic pressure,
1.
DYNA
Figure 4: Spectral Response of the fiat plate corresponding to Fig 3. Central
Node for 12Pa rms pressure, DYNA calculation.
3
2.5
2
1
o
^ 1
0.5
0
Figure 5: Rms Central Deflection of the plate versus sound pressure level
Comparison between DYNA results and linear theory.
3
2.5
2
■£
£
E
I
I
o
1
0.5
0
Figure 6: Central deflection of the plate versus pressure, NASTRAN calcula¬
tions compared to linear theory.
500 1000 1500 2000 2500 3000 3500
Static Pressure (Pa)
rms Pressure (Pa)
1022
Figure 7: Variation of fundamental frequency of the (1,1) mode with rms
pressure, Comparison between DYNA and theory.
Damping 2%; Stoctiastic Fn #1 ; 3Sx29 Nodes
0 - 1 - 1 - 1 - 1 - -
0 200 400 600 800 1000 1200
rms Pressure (Pa)
130 140 145 150 153 155
SPL (dB)
Figure 8: Variation of width of the (1,1) mode resonance peak with rms pres¬
sure, Comparison between DYNA and theory.
1023
Figure 9: DYNA model predictions for random vibration of the plate with
semi-clamped boundary conditions.
Figure 10: Random Vibration results with compressive load in the y-direction
of -3.46N/mm and SPL of l.OPa.
Figure 11: Random Acoustic Loading of SPL=12Pa superimposed on a static
pressure of TOOPa.
1024
Figure 12: Random acoustic loading of SPL=700Pa superimposed on a thermal
load of 10 deg, clamped edges. — No Snap Through
Figure 13: Random acoustic loading of SPL— 2kPa superimposed on a thermal
load of 10 deg, clamped edges. — Nearly continuous snap-through.
Figure 14: Random acoustic loading of SPL=4kPa superimposed on a thermal
load of 10 deg, clamped edges. — Dominant acoustic load.
1026
SYSTEM IDENTIFICATION II
ROBUST SUBSYSTEM ESTIMATION
USING ARMA-MODELLING
IN THE FREQUENCY DOMAIN
by U. Prells, A. W. Lees, M. 1. Friswell and M. G. Smart,
Department of Mechanical Engineering of the University of Wales
Swansea,
Singleton Park, Swansea SA2 8PP, United Kingdom
ABSTRACT
This paper reflects early results of the research on modelling the influence of
the foundation on the dynamics of the rotor. The foundation is connected to
the rotor via journal bearings. Dynamic models exist for the subsystems of
the rotor and of the bearings; the first is reliable but the latter is uncertain.
The foundation model is unknown and has to be estimated using rundown
data.These are measured responses of the foundation at the bearings due
to unbalance forces of the rotor which are assumed to be known. Uncer¬
tainties in the bearing model will be transfered to the estimated foundation
parameters. The main scope of this paper is to introduce a method which
enables the decoupling of the problem of model estimation and the problem
of the influence of the bearing model uncertainty.
The influence of changes in the model of the bearings on the estimation
of the foundation model is mainly due to the sensitivity of the computed
forces applied to the foundation at the bearings. These are used together
with the associated measured responses to estimate the foundation model in
the frequency domain. Using an ARMA model in the frequency domain it is
possible to estimate a filtered foundation model rather than the foundation
model itself. The filter is defined in such a way that the resulting force has
minimum sensitivity with respect to deviations in the model of the bearings.
This leads to a robust estimation of the filtered model of the foundation.
Since the filter can be defined in terms of the models of the rotor and of the
bearings only, the problems of estimating the foundation’s influence and of
the sensitivity of the estimates with respect to the model of the bearings
are decoupled.
The method is demonstrated by a simple example of a single-shaft rotor.
Even if the errors in the bearing model are about 50 % the relative input and
output errors of the filtered foundation model are of the same magnitude
as the round-off and truncating errors.
1 INTRODUCTION
An important part of a machine monitoring system for fault diagnostics of
1027
a turbo generator is a reliable mathematical model. This model includes
the subsystems of the rotor, the bearings and the foundation. The model of
the rotor represents the most reliable knowledge, the model of the journal
bearings is uncertain, and despite of intensive research it is not yet possible
to define a model for the foundation which refiects the dynamical contribu¬
tion to the rotor with sufficient accuracy. The first step to determine the
contribution of the foundation on the rotor’s dynamic performance is the
estimation of a reliable foundation model.
Rundown data are available, i.e. displacements ufb{<->^) ^ of the
foundation at the bearings which are due to an unbalance force /c/(a;) G
of the m-shaft rotor, given at discrete frequencies a; € := {wi, ■ • • , wm}?
and this data may be used to estimate the foundation model. A com¬
mon method ([1],[2],[3],[4],[5]) is to estimate the unknown dynamic stiffness
matrix 6 i^4mx4m foundation at the bearings using the in¬
put/output equation
F{uj)upb{(^) = (1)
where the force fpB of the foundation at the bearings can be expressed
by dynamic condensation in terms of the data ups, fu and in terms of the
dynamic stiffness matrices Ar,B of the rotor and the bearings respectively
yielding
fpB = -Bufb + [0, B]A~^q ^ ^ . (2)
Here the dynamic stiffness matrix Arb of the rotor mounted on the bearings
is partitioned with respect to the n inner degrees-of-freedom (dof) of the
rotor and to the 4m connecting (interface) dof
^RB
Aru Arib
Arbi Arbb + B
(3)
The non-zero components of the force fpi G (D” in eq. (2) of the inner part
of the rotor are the components of the unbalance force /y, i.e. introducing
the control matrix Su € ^ dynamic stiffness
matrix Ar of the rotor is given in terms of the matrices of inertia and
stiffness which are defined by modal analysis and by its physical data given
by the manufacturer. Each of the m shafts of the rotor is connected to
the foundation usually via 2 journal bearings. Since the dynamic stiffness
matrix B of the journal bearings represents a model for the oil film it consists
of connecting dof only. It can be shown that B is block diagonal
B =
0
0
(4)
containing the dynamic stiffness matrices Bi = Ki+ jujDi.i = 1, • • • ,m, of
the m bearings. Ki, Di are the matrices of stiffness and damping respectively
1028
which result from linearisation and are in general non-symmetric and non¬
singular. Eq. (1) is then used to estimate the foundation transfer function
F{u). This has been discussed in several papers ([1],[2],[3],[4],[5]). Lees et
al. [3] pointed out that fpB is sensitive with respect to deviations in the
model of the bearings over part of the frequency range. This sensitivity is
transfered to the model estimates.
In this paper a method is introduced which enables the decoupling of the
two problems of model estimation and of sensitivity of the foundation model
with respect to the model of the bearings. The basic idea of this method is
to estimate a transfer function H{u) which maps the displacements ufb{^)
to a force /^(w) rather than the force i-e.
H(u)ufb{^) = (5)
In extension of the earlier method the force //f(a;) can be chosen to be of
minimum sensitivity with respect to the model of the bearings. This robust
estimated transfer function H{uj) is related to that of the foundation F(uj)
by a transformation P{u)
H{lo) = PMF(a;), (6)
which of course retains the sensitivity with respect to the model of the
bearings. But since P{uj) only depends on the models of the rotor and the
bearings in the case of a modification within the model of the bearings no
new model estimation has to be performed because this has been done ro¬
bustly with respect to such model changes.
2 THE OPTIMUM CHOICE OF THE
FORCE VECTOR
As stated in Lees et al. [3] the sensitivity of the force /fb with respect to B
is mainly due to the inversion of the matrix Arb in eq. (2). It can be shown
that the condensation method of estimating the force /fb results from the
special case of eliminating the last 4m rows of the matrix
W :=
Arii
^RBI
0
Arib
ArbB -b P
^ ^(n+8m.)x(n+4m)
(7)
which can be written as Arb = T'^W ^ (l^(n+4m.)x(n+4m) defining the
selecting matrix of the master dof as
T := [ei, • • • , € R(-+8n)x(n+4m)^ (S)
1029
where in general en denotes a unit vector of appropriate dimension contain¬
ing zeros everywhere but in the nth place. In extension to the force fpB
defined in eq. (2) for an arbitrary selecting matrix T € the
condensation leads to a force fn given by
j-^LT
In+Sm-W{T'^Wy'T^
” *■ V*' . . .
=:P
(
Bufb
V }
( ^ \
0
V -^4m J
fpB-
(9)
Here T-^ € denotes the matrix which selects the slave dof,
and in general In denotes the unit matrix of dimension n. Indeed, inserting
the special choice of T from eq. (8) into eq. (9) leads to the sensitive force
Ih = fpB as defined in eq, (2).
The reason for the sensitivity of /fb is that the subsystem of the rotor
has low damping. Near the resonance frequencies of Arb its large condition
number depends sensitively on B. Thus the sensitivity of Jfb with respect
to B is due to a large condition number of Arb • Let T denote the set of all
possible selecting matrices, i.e.
r := {[ei..--',ew4„]:e4elR"+*“.
l<4<n + 8m, V /c = 1, • • « ,n + 4m}, (10)
One criterion for an optimum choice of the force Jr may be formulated as
the following minimisation problem:
Criterion 1:
The optimum choice is the solution of
mmcond{T^W), (11)
where W is defined in eg. (7).
A low condition number is necessary but not sufficient in order to provide
a low sensitivity of the force /h* Therefore a numerical test can be applied
using stochastic deviations in the bearing model. Let A A consist of
uniform distributed non-correlated random numbers with zero mean values
and variances equal to 1/3 for alH G 1, ■ • * , 2m. Define
ABi = ABfisi, Ti) := siAKi -h A A, (12)
where the positive scalars si.Ti control the magnitude of the random error
of the i-th bearing model. Thus, the error AB = AB{s,r) of the bear¬
ing model is well defined for s := (si, • ■ • , 52771)"^ and r := (ri, • • • ,r2mV ■
1030
Regard the force fn = f{uj,T,AB) as a function of the selecting ma¬
trix T and the bearing model error AB. For I random samples AB{k) =
AB(r{k),s{k)),k == calculate for each frequency a; G the up¬
per and lower bounds for the real and imaginary part of each component
/i, ^ = 1, • • • , 4m of the force vector /, i.e.:
//Lax(“.U
;= max Re {/i(a;,T, AB(/c))} ,
(13)
:= min Re {fi{u,T, AB{k))} ,
(14)
:= ^max^Im {/i(a;,T, AB(/c))} ,
(15)
fLni^.T)
:= min Im AB{k))} .
k=l,"-,l
(16)
Defining the force vectors
/max(w,T)
■■= T)+j- T),
(17)
/mm(w,T)
(18)
the second criterion can be formulated as a minimax problem:
Criterion 2:
The optimum selection is obtained from
4m
minmax^l/imax(w,r) - (19)
TST uj£u . -
2 = 1
Before the method outlined is demonstrated by an example some aspects of
the mathematical model of the foundation and methods for its estimation
based on the input/output equation (5) will now be considered.
3 ESTIMATION OF THE FOUNDATION
MODEL
The purpose of this section is to estimate the unknown foundation model
represented by the matrix
F{uj) = Afbb{^) - Afbi{^)Af]j{uj)Afib{^)‘ (20)
This expression results from dynamic condensation of the dynamic stiffness
matrix of the foundation
Af
Afbb Afib
Afib Afh
(21)
1031
which is partitioned with respect to its inner dof (index I) and those dof
coupled to the bearings (index B). For viscous damped linear elastomechan-
ical models the dynamic stiffness matrix Aj?(a;) of the foundation is given
by ^
Af(uj) := (22)
i=0
The matrices Ai are real valued and represent the contributions of stijffness,
damping and inertia for z = 0, 1, 2 respectively. In this case the identifica¬
tion of the foundation model requires the estimation of the three matrices
Ai which are parameterised by introducing dimensionless adjustment pa¬
rameters aik € IR, for all /c = 1, * • • , ^ = 0, 1, 2 (see for instance [6] or
[7] ). Those parameters are related to given real- valued matrices Sik by
Ni
Ai{ai)
Jk=i
Writing the adjustment parameters as one vector a"’’ ;= (af,aj’,aj) G IR^,
p := ATq + Ni + W2, the estimation of the foundation model is equivalent to
the estimation of the parameter vector a. The dynamic stiffness matrix of
the foundation becomes a nonlinear function of this parameter vector
^"(0;) =F[(jj^a) — Afbb[^-)0) “ Afb7(ci;j a)Ap}j(a;, a)Af’/5(a;, a). (24)
Substituting the measured quantities for upB and fu into eqs. (5) and
(9) the parameter vector a is usually estimated by minimising some norm of
the difference between measured and calculated quantities, called residuals
[8] . Using equation (5) is equivalent to the input residual method. Defining
the ith partial input residual as
(25)
where the dependency on the model parameters a of the input vector is
defined by
f{u), a) := P{uj)F[u), a)uFB(w), (26)
the cost function to be minimised is given by
M
JKa):=£^;}(i)W,(i)^/(i), (27)
i=l
where Wj{i) represents a weighting matrix for the zth partial residual and
the superscript f denotes the conjugate- transpose. The inverse problem (27)
is nonlinear with respect to the parameters to be estimated. Thus, there
is no advantage relative to the output residual method. Defining the ith
partial output residual as
vo{i) := u{ui, a) - (28)
1032
where the model output is defined by
(29)
0
0
ApiiiuJ^a) .
(30)
(31)
Woii) denotes a weighting matrix for the ith partial output residual.
Mathematical modelling is always purpose orientated [9]. In the case
discussed in this paper the purpose is to estimate the influence of the foun¬
dation on the dynamics of the rotor. For this purpose, no physically inter¬
pretable model is necessary in order to model this influence. In the next
section an alternative mathematical model is introduced which leads to a
linear inverse problem.
3.1 ARMA MODELLING IN THE FREQUENCY
DOMAIN: THE FILTER MODEL
Auto Regressive Moving Average models are well developed (see for in¬
stance [10], [7]) in order to simulate dynamic system behaviour. ARMA
models are defined in the time domain by
- lAt) = - ^A^), (32)
i=0 i=0
where the present output (state or displacement) u{t) due to the present
input f{t) depends on rio past outputs and on rii past inputs.
In the frequency domain eq. (32) leads to a (frequency-) filter model [7].
With reference to eq. (5) it has the form
Su
0
0
0
u(w,a) := [0,0,/2(m+i),0].4 ^(w,a)
with the dynamic stiffness matrix A of the entire model
A{u, a) =
Arii{u) Arib{^^) 0
Arbi{^) Arbb{‘^) B{lo) —B(u)
0 —B{uj) R(a;) 4- Afbs(^) g)
0 0 Afib{^jO,)
the cost function to be minimised is
M
Jo(a) :=E^oW^oW«o(i).
The output and input powers Uo^rii respectively, and the matrices
(•SA:)fc=o,-,ni are called filter^parameters and Jiave to be esti¬
mated. Of course the minimum of det[A(a;)] and of det[B(cj)] correspond
to the resonance and anti-resonance frequencies of the subsystem of the
foundation respectively.
For an optimum choice of P (see eq. (9)) the estimation of A and B can
be considered to be independent of the precise values of the model of the
bearings. Thus, the problem of the uncertainty in the bearmg models and
the problem of model estimation are decoupled. If A and B are estimated
refering to eq. (5) then
= H. (34)
The estimation of the filter parameters is robust with respect to deviations
in the bearings model. Thus, the uncertainty of the estimation of the foun¬
dation model F is due to the inversion of the matrix P only
F = P-^H = p-'-B~^A, (35)
which represents a problem a priori and which occurs only in the calculation
of the force of the foundation at the bearings
/PB = P-^B~^Aufb- (36)
Of course the force vector fpB is sensitive to changes in the bearings model
but only due to corresponding changes in P. The estimated part B A is
robust with respect to changes in the bearings model.
In order to calculate the response ufb iio explicit calculation of the
inverse of P is necessary,
upB = A Bfn- (37)
Since the estimated model and the force Jh are insensitive with respect to
the bearings model the estimation otupB is robust in this sense.
Of course the influence of errors in up b and fu have not yet been taken
into account. Accordingly the model powers n^n^, must be estimated as
well as the matrices Ai.Bk- The estimation method is outlined in the fol¬
lowing section.
3.2 ESTIMATION OF THE FILTER
PARAMETERS
In order to estimate the filter parameters the least squares method can be
applied to minimise the equation error in eq. (33). Defining the zth partial
equation residual as
vsij) '= A[ui)upB[^i) - B{uJi)fH{^i) (38)
1034
the cost function to be minimized is given by
M
i=l
(39)
where W^ii) denotes a weighting matrix for the ith. equation residual. As¬
suming WeIi) = Am for alH = 1, • • ’ , M, the filter equation (33) can be
extended for M excitation frequencies as
= (40)
where C/, 2^ and A are defined by
U
Z
A
• • • 5 ,
r a;i 0 1
lom
(41)
(42)
(43)
The solution of the minimisation problem (39) is equivalent to the normal
solution of eq. (40) which can be rewritten as
[Afio 1 ' ' ' 1 -Ao ) ‘ : ■^o]
=: V
U
Z
= 0
(44)
Because the filter parameters represented by the matrix V €
j^4mx4m(no+ni+2) leal-valued, equation (44) must be satisfied for the
real and imaginary parts of the matrix Y € which finally
yields
1/ [Re {Y} , Im {Y}] =: VA = 0. (45)
This problem does not lead to a unique solution for the filter parameters.
Indeed, for any arbitrary non-singular matrix C
CAufb = (46)
is also a solution. But since one is interested (see eq. (34)) in the product
B~^ A (or its inverse) only this final result is of interest and this product is
unique.
1035
As a necessary and a sufficient condition for a full-rank solution V of
eq. (45) the matrix X € 5^4m(no+ni4-2)x2M a rank deficiency of
4m, i.e.
rank(A) = 4m (no + n^ + 1). (47)
Of course this problem has to be treated numerically. The rank decision
is usually made by looking to the singular values 7(no, rii) 6 5^4m(no+ni+2)
of the matrix X = X{no,ni). Because one cannot expect to achieve zero
rather than relative small singular values one has to define a cut-off limit.
This is due to the fact that the equation error (39) can be made arbitrary
small by increasing the degree p := no + Ui of the filter model. The same
situation occurs if one looks at the maximum relative input error
6/ :=
max
(48)
or to the maximum relative output error
en :== max
II^fs(^2)1I
(49)
With increasing degree p the errors ej and cq can be made arbitrary small.
This is a typical expression for an ill-posed problem which can be turned
into an well-posed problem by applying regularisation methods [11]. To
choose an appropriate regularisation method needs further investigation and
is beyond the scope of this paper. In the next section the method of choosing
the optimum force vector fn is demonstrated by a simple example.
4 A SIMPLE EXAMPLE
The test model is depicted in Fig. 1. The one-shaft-rotor is simulated by
an Euler-Bernoulli beam which is spatially discretised with 10 dof. Accord¬
ing to the partition with respect to inner points and interface points (see
eq. (3)) the number n of inner rotor dof is 8 and the number of connecting
dof is 2. Only one translation dof of the rotor is connected to each bearing
which are modelled by massless springs with stiffnesses ki — 1.77 • 10® and
k2 = 3-54 • 10® N/m respectively. The foundation is modelled by an uncon¬
nected pair of masses mi = 90, m2 = 135 kg and springs with stiffnesses
kfi = /c/2 = 1.77 • 10® N/m. The force fu due to an unbalance 6 = 0.01
kg-m is given by fu{<^) ■= € IR. The force vector f^i e IR® is assumed
to have one non-zero component only, i.e. Jri fuc^. The frequency range
between 0 and 250 Hz is discretised with equally spaced stepsize of 0.5 Hz.
The selecting matrix T € of the master dof is assumed to consist of
the unit vector 64 in order to select out the unbalance force fjj because this
1036
Figure 1: The simple test model
excitation is independent of the model of the bearings and therefore of min¬
imum sensitivity. Thus the remaining redundancy consists in eliminating
one row of the matrix
G := [ei, 82. 83. 65. • • • . eul'' W € (50)
The result of the first criterion are depicted in Fig. 2. It shows the frequency
dependent condition numbers for the elimination of each row of the matrix
G in turn. This leads to an optimum choice by eliminating the 4-th row of
G. Thus, the optimum choice of the master dof is given by T*^ = [64,65].
This result is confirmed by applying the second optimisation criterion.
For this purpose a uniform distributed uncorrelated random error with
zero mean value is added to the stiffnesses of the bearings simulating a
model variation of 50 per cent, i.e. ki — ^ ki -1- Nzikil2, where =
1,2, are uncorrelated random numbers with expectation value E{Aki} = 0
and with variance E{AkiAkj} = For a size of I - 500 random
samples, the upper and lower bounds /max, /min and /f max, /h min of the
force fpB = /fb(<^, A5) and of the force Jh = /(w,T, AB) respectively
with the selecting matrix T - [64,65], have been calculated. In contrast to
the maximum difference of upper and lower bounds of the force f^B of ~ 20,
that of the force fn is of the order of the computational accuracy ~ 10"'^,
and is therefore negligible. In a first step the force /fb is used to estimate
the filter parameters of the model F of the foundation. Solving the singular
value decomposition for all input and output powers {ni.rio) 6 [0, 5]^ C
the calculation of the maximum relative input and of the maximum output
errors as defined by eqs. (48) and (49) with fn ~ Ifb leads to the results
1037
Elimination of row number...
Frequency [Hz]
Frequency [Hz]
Figure 2: Frequency dependent condition numbers
depicted in Fig. 3. For a model realisation with [ni.rio) = (0,2) the values
of the maximum relative errors are approximately ej 7.1 • and eo ~
9.0-10“”^ which corresponds to the computational accuracy. Using this model
the estimates of the filter matrices correspond within the computational
accuracy to those of the ‘true’ foundation model.
Using variations of the force fpB between the bounds /max, /min froni
the second criterion the associated upper and lower bounds of the relative
input and output error have been calculated. The influence of the variation
1038
Firgure 3: Maximum relative input and output error for different
input powers rii and output powers rio using the force vector fpB
of the bearing stiffness of 50% leads via the associated variation of the
force fpB to drastic variations of the relative input and output errors. The
difference of upper and lower bound of the maximum relative input error is
1039
Output power
Figure 4: Maximum relative input and output error for different
input powers rii and output powers Uo using the optimised force
vector Jh
of order 100 and that of the maximum relative output error is approximately
LOl
The situation is different using the optimised force fn in order to es¬
timate the filter parameters of the model H. For each input and output
power {ni,no) e [0,25]^ C the maximum relative input error and the
maximum relative output error have been calculated. The result is shown in
Fig. 4. For a maximum relative input error e; 6.5 • 10“® a filter model of
degree 12 is available with the powers (n^, = (4, 8). This model produces
a maximum relative output error eo ^ 10“®.
Analogous to the robustness investigations for fpB now for the force
/h the upper and lower bounds of the relative input and of the relative
output error due to the random variation in the bearing models have been
calculated. For the chosen model with powers (nj,no) = (4, 8) the difference
of upper and lower bound of the relative input error as well that of the
relative output error are of about the same order 10“^. Thus, compared
with the order of variation 100 and 10"^ of the direct foundation model
estimate the estimate of H is robust with respect to changes in the bearing
model.
5 CONCLUSION
In this paper a method is introduced which enables the decoupling of the
two problems of model estimation and of sensitivity of the foundation model
with respect to the model of the bearings. The method produces an opti¬
mised choice of the input /output equation which provides a transfer func¬
tion estimation that is robust with respect to deviations in the model of
the bearings. For the foundation model estimation a filter model is intro¬
duced. This modelling strategy has the advantage of leading to a linear
inverse problem. The disadvantage is that with increasing model degree
the equation error can be made arbitrarily small. Because this error should
not become smaller than the accuracy of the data, a cut-off limit has to be
determined a priori. Further investigations should allow the cut-off limit to
be related to the data errors.
REFERENCES
1. Feng, N.S. and Hahn, E.J., Including Foundation Effects on the Vi¬
bration Behaviour of Rotating Machinery. Mechanical Systems and
Signal Processing, 1995 Vol. 9, No. 3, pp. 243-256.
2. Friswell, M.I., Lees, A.W. and Smart, M.G., Model Updating Tech¬
niques Applied to Turbo-Generators Mounted on Flexible Founda¬
tions. NAFEMS Second International Conference: Structural Dynam¬
ics Modelling Test, Analysis and Correlation, Glasgow: NAFEMS,
1996 pp. 461-472.
1041
3. Lees, A.W. and Friswell, M.L, Estimation of Forces Exerted on Ma¬
chine Foundations. Identification in Engineering Systems, Wiltshire;
The Cromwell Press Ltd., 1996, pp. 793-803.
4. Smart, M.G,, Friswell, M.L, Lees, A.W. and Prells, U., Errors in
estimating turbo-generator foundation parameters. In Proceedings
ISMA21 - Noise and Vibration Engineering, ed. P. Sas, Katholieke
Universiteit Leuven, Belgium, 1996, Vol. II, pp. 1225-1235
5. Zanetta, G.A., Identification Methods in the Dynamics of Turbogener¬
ator Rotors. The International Conference on Vibrations in Rotating
Machinery, IMechE, C432/092, 1992, pp. 173-181.
6. Friswell, M.L and Mottershead, J.E., Finite Element Model Updating
in Structural Dynamics. Dordrecht, Boston, London: Kluwer Aca¬
demic Publishers, 1995.
7. Natke, H.G., Einfuhrung in die Theorie und Praxis der Zeitreihen-
und Modalanalyse - Identifikation schwingungsfdhiger elastomechanis-
cher Systeme. Braunschweig, Wiesbaden: Friedrich Vieweg Sz Sohn,
1993.
8. Natke, H.G., Lallement, G., Cottin, N. and Prells, U., Properties of
Various Residuals within Updating of Mathematical Models. Inverse
Problems in Engineering, Vol. 1, 1995, pp. 329-348.
9. Natke, H.G., What is a true mathematical model? - A discussion of
system and model definitions. Inverse Problems in Engineering, 1995,
Vol. 1, pp. 267-272.
10. Gawronski, W. and Natke, H.G., On ARMA Models for Vibrating
Systems. Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 3,
pp. 150-156.
11. Baumeister,J., Stable Solution of Inverse Problems. Braunschweig,
Wiesbaden: Friedrich Vieweg &; Sohn, 1987.
ACKNOWLEDGEMENTS
The authors are indebted to Nuclear Electric Ltd and Magnox Pic for fund¬
ing the research project BB/G/40068/A to develop methods which enables
the estimation of the influence of the foundation on the dynamics of the
rotor. Dr. Friswell gratefully acknowledges the support of the Engineering
and Physical Sciences Research Council through the award of an Advanced
Fellowship.
1042
MATHEMATICAL HYSTERESIS MODELS AND
THEIR APPLICATION TO NONLINEAR
ISOLATION SYSTEMS
Y.Q. Ni, J.M. Ko and C.W. Wong
Department of Civil and Structural Engineering
The Hong Kong Polytechnic University, Hong, Kong
Abstract
Two mathematical hysteresis models, the Duhem-Madelung (DM)
model and the Preisach model, are introduced to represent the
hysteretic behavior inherent in nonlinear damping devices. The DM
model generates the hysteresis with local memory. Making use of the
Duhem operator, the constitutive relation can be described by single¬
valued functions with two variables in transformed state variable
spaces. This makes it feasible to apply the force-surface nonparametric
identification technique to hysteretic systems. The Preisach model can
represent the hysteresis with nonlocal memory. It is particularly
suitable for describing the selective-memory hysteresis which appears
in some friction-type isolators. An accurate frequency-domain method
is developed for analyzing the periodic forced vibration of hysteretic
isolation systems defined by these models. A case study of wire-cable
vibration isolator is illustrated.
1 . Introduction
The dynamic response of a structure is highly dependent on
the ability of its members and connections to dissipate energy by
means of hysteretic behavior. The assessment of this behavior
can be done by means of experimental tests and the use of
analytical models that take into account the main characteristics
of this nonlinear mechanism. Although a variety of hysteresis
models have been proposed in the past decades, many structural
systems exhibit more complicated hysteretic performance (mainly
due to stiffness or/ and strength degrading) which the models in
existence are reluctant and even inapplicable to depict [1,2]. On
the other hand, nonlinear vibration isolation has recently been
recognized as one of effective vibration control techniques. In
1043
particular, hysteretic isolation devices have got wide applications
owing to their good diy friction damping performance. These
hysteretic isolators may exhibit very complicated features such
as asymmetric hysteresis, soft-hardening hysteresis, nonlocal
selective-memory hysteresis [3-5]. None of the models available
currently in structural and mechanical areas can represent all
these hysteresis characteristics. Other more elaborate hysteresis
models need to be established for this purpose.
In reality, hysteresis phenomenon occurs in many different
areas of science, and has been attracting the attention of many
investigators for a long time. However, the true meaning of
hysteresis varies from one area to another due to lack of a
stringent mathematical definition of hysteresis. Fortunately,
because of the applicative interest and obvious importance of
hysteresis phenomenon, Russian mathematicians in 1970’s and
the Western mathematicians in 1980’s, began to study hysteresis
systematically as a new field of mathematical research [6,7]. They
also deal with the hysteresis models proposed by physicists and
engineers in various areas, but they separate these models from
their physical meanings and formulate them in a purely
mathematical form by introducing the concept of hysteresis
operators. Such mathematical exposition and treatment can
generalize a specific model from a particular area as a general
mathematical model which is applicable to the description of
hysteresis in other areas. In this paper, two mathematical
hysteresis models are introduced and the related problems such
as identification and response analysis encountered in their
application to nonlinear isolation systems are addressed.
2. Definition of Hysteresis
Hysteresis loops give the most direct indication of hysteresis
phenomena. But it is intended here to introduce a mathematical
definition of hysteresis. Let us consider a constitutive law: u r,
which relates an input variable u(t) and an output variable r\t).
For a structural or mechanical system, u(t) denotes displacement
(strain); r{t) represents restoring force (stress); t is time. We can
define hysteresis as a special type of memoiy-based relation
1044
between u(t) and r(t). It appears when the output r[t) is not
uniquely determined by the input u(t) at the same instant t, but
instead r(t) depends on the evolution of u in the interval [0, t] and
possibly also on the initial value ro, i.e.
r(t) = iR[u(-),ro](t) (1)
where the memory-based functional iR[u(-),ro](t) is referred to as
hysteresis operator. In order to exclude viscosity-type memory
such as those represented by time convolution, we require that 91
is rate-independent, i.e. that r(t) depends just on the range of u in
[0, t] and on the order in which values have been attained, not on
its velocity. In reality, memory effects may be not purely rate
independent as hysteresis is coupled with viscosity- type effects.
However, as shown later, in most cases the rate independent
feature of hysteresis is consistent with experimental findings,
especially when evolution (variation in time) is not too fast.
3. Duhem-Madelung (DM) Model
3.1 Formulation
The DM model can be defined with or without referring to a
confined hysteresis region. For the structural or mechanical
hysteretic systems, it is not necessary to introduce the notion of
bounded curves because there exists neither the saturation state
nor the major loop. In this instance, the DM model establishes a
mapping (named Duhem operator) 91: (u, ro) rby postulating the
following Cauchy problem [7,8]
= g, (li. '■) ■ w, (t) - 02 (u, r) • u_ (t) (2a)
r(0) = r„ (2b)
where an overdot denotes the derivative with respect to t; gi{u,r]
and g2{ii,r) are referred to as ascending and descending functions
(curves) respectively; and
tijt) = max[0,u(t)] = t [|u(t)| + u(t)] (3a)
u.(t) = min[0,u(t)] = ^[|u(t)| - u(f)] (3b)
Eq. (2a) can be rewritten as
f(f) = p[u,r,sgn(u)]-u(t) (4)
1045
u> 0
u<0
(5)
in which the describing function has the form
g{u,r,sgn{u)]
fg{u,r,l) = g,{u,r)
[g[u,r-l) = g^(u,r)
It is obvious that the DM model is rate independent. In
addition, it is specially noted that in this constitutive law the
output r{t) is not directly dependent on the entire history of u{t)
through [0,t]; but instead depends only on the local histoiy
covered since the last change of sgn(u) and on the value of the
output at this switching instant. It means that the output can
only change its character when the input changes direction. As a
consequence, the DM model usually represents the hysteresis
with local memory except that the functions gi(u,?) and g2{u,7) are
re-specified as hysteresis operators.
Within the framework of DM formulation, the ascending and
descending functions gi(u,T] and g2(u,r) are just required to fulfill
suitable regularity conditions and need not to be specified in
specific expressions, so both the form and parameters of the
functions can be fine-tuned to match experimental findings. On
the other hand, the DM formulation can deduce a wide kind of
differential-type hysteresis models such as Bouc-Wen model,
Ozdemir’s model, Yar-Hammond bilinear model and Dahfs
frictional model. For the Bouc-Wen model
r(t) = K.u{t) + z(t) (6a)
z{t) = au(t] - P|ti(f)|z(f)|z(t)|'”‘ - YU(t)|z(f )|" (6b)
it corresponds to the DM model with the specific ascending and
descending functions as
grj(u,r) = a + K “ [y + p sgn(r -Ku)]|r -Kup (7a)
g2(w,r) = a + K - [y - p sgn(r -Kw)]|r -ku|” (7b)
and for the Yar-Hammond bilinear model
f{t) = {a - y sgn(ii) sgn[r - p sgn(ii)]}ii (8)
its describing function is independent of u(t) as follows
5f[u,r,sgn(u)l = gf[r,sgn(u)l = a -y sgn(u)sgn[r - p sgn(u)] (9)
Hence, the Duhem operator also provides an accessible way
to construct novel hysteresis models by prescribing specific
1046
expressions of the ascending and descending functions.
Following this approach, it is possible to formulate some models
which allow the description of special hysteretic characteristics
observed in experiments, such as soft-hardening hysteresis,
hardening hysteresis with overlapping loading envelope, and
asymmetric hysteresis [5].
3.2 Identification
System identification techniques are classified as parametric
and nonparametric procedures. The parametric identification
requires that the structure of system model is a priori known. The
advantage of nonparametric identification methods is that they
do not require a priori the knowledge of system model. The most
used nonparametric procedure for nonlinear systems is the force
mapping (or called force surface) method [9]. This method is
based on the use of polynomial approximation of nonlinear
restoring force in terms of two variables — ^the displacement u(t)
and the velocity u{t) . For nonlinear hysteretic systems, however,
the hysteretic restoring force appears as a multivalued function
with respect to the variables u{t) and u(t) due to its history-
dependent and non-holonomic nature. This renders the force
mapping method inapplicable to hysteretic systems, although
some efforts have been made to reduce the multivaluedness of
the force surface [10,11].
One of the appealing virtues of the DM model is that it can
circumvent this difficulty. Making use of the Duhem operator,
the hysteretic constitutive relation of Eq.(l) is described by two
continuous, single-valued functions gi(u,r) and g2{u,r) in terms of
the displacement u(t) and the restoring force r{t). Thus, single¬
valued “force” surfaces gi{u,r) and g2{u,7) can be formulated in
the subspaces of the state variables {u,r,gi) and (u,r,g2), and can
be identified by using the force mapping technique. Following
this formulation, a nonparametric identification method is
developed by the authors [12]. In this method, the functions
g\(u,7) and g2{u,7) are expressed in terms of shifted generalized
orthogonal polynomials with respect to u and r as follows
gM,r) = i = 0’'(u)G<''0(r) (10a)
i=lj=l
1047
(10b)
g^r] = z i(|>,.(r)gf(i>,(u) = 0^(u)G'^'<l>(r)
I=lj=l
where = [gf ]^xn =[5'zf]mxn ^re called the expansion-
coefficient matrices of gi{u,r) and g2{u,T]. Some algorithms have
been proposed to estimate the values of these coefficient matrices
based on experimentally observed input and output data. It
should be noted that here the vectors 0(u) and <^{r) are shifted
generalized orthogonal polynomials [13]. They are formulated on
the basis of common recurrence relations and orthogonal rule,
and cover all kinds of individual orthogonal polynomials as well
as non- orthogonal Taylor series. Consequently, they can obtain
specific polynomial-approximation solutions of the same. problem
in terms of Chebyshev, Legendre, Laguerre, Jacobi, Hermite and
Ultraspherical polynomials and Taylor series as special cases.
4. Preisach Model
4. 1 Formulation
The intent of introducing the Preisach model is to supply the
lack of a suitable hysteresis model in structural and mechanical
areas, which is both capable of representing nonlocal hysteresis
and mathematically tractable. Experiments revealed that the
hysteretic restoring force of some cable-type vibration isolators
relates mainly to the peak displacements incurred by them in the
past deformation [3]. It will be shown that the Preisach model is
especially effective in representing such nonlocal but selective-
memory hysteresis, in which only some past input extrema (not
the entire input variations) leave their marks upon future states
of hysteresis nonlinearities.
The Preisach model is constructed as a superposition of a
continuous family of elementary rectangular loops, called relay
hysteresis operators as shown in Fig. 1. That is [7,14],
r(f) = 'R[u(-)](f) = j||^(a,P)y„D[u(t)]dadf5 (11)
a>P
where ia(a,|3) > 0 is a weight function, usually with support on a
bounded set in the (a,p)-plane, named Preisach plane; Ya.p[^(^)] is
the relay hysteresis operator with thresholds a > p. Outputs of
1048
these simplest hysteresis operators have only two values +1 and
-1, so can be interpreted as two-position relays with “up” and
“down” positions corresponding to ya,p[^^(^)l=+l and
1
+1
p
a
-1
Fig. 1 Relay Hysteresis Operator
Hence, the Preisach model of Eq.(ll) can be interpreted as a
spectral decomposition of the complicated hysteresis operator iR,
that usually has nonlocal memory, into the simplest hysteresis
operators 7ct,p with local memory. In the following, we illustrate
how the model depicts the nonlocal selective-memory feature.
Consider a triangle T in the half-plane a > (3 as shown in
Fig.2. It is assumed that the weight function )i(a,p) is confined in
the triangle T, i.e. }i(a,p) is equal to zero outside T. Following the
Preisach formulation, at any time instant t, the triangle T can be
subdivided into two sets: S^(t) consisting of points (a,P) for which
the corresponding Yc^.p-operators are in the “up” position; and S^(t)
consisting of points {a,P) for which the corresponding Ya.p“
operators are in the “down” position. The interface L(t) between
S^(t) and Sr(t) is a staircase line whose vertices have a and p
coordinates coinciding respectively with local maxima and
minima of input at previous instants of time. The nonlocal
selective-memory is stored in this way. Thus, the output r(^) a-t
any instant t can be expressed equivalently as [14]
r{t)= J|^(a,P)dadp- |jM,(a,P)dadp (12)
S'(£) S'(t)
It should be noted that the Preisach model does not
accumulate all past extremum values of input. Some of them can
be wiped out by sequent input variations following the wiping-out
property {deletion rule): each local input maximum wipes out the
1049
vertices of L(t) whose a-coordinates are below this maximum, and
each local minimum wipes out the vertices whose p-coordinates
are above this minimum. In other words, only the alternating
series of dominant input extrema are stored by the Preisach
model; all other input extrema are erased.
Fig. 2 Input Sequence and Preisach Plane
4.2 Identification
It is seen from Eq.(l 1) that the Preisach model is governed by
the weight function |i(a,p) after determining L(t) which depends
on the input sequence. !i(a,P) is a single-valued function with
respect to two variables a and p. Hence, the aforementioned
nonparametric identification method can be also implemented to
identify iLL(a,p) by expanding it in a similar expression to Eq.(lO).
An alternating approach is to define the following function
H'(a',P') = |j|a(a,p)dadp = ^.[^ ^(ot,P)da]dp (13)
r(cc'.3')
where T(a',p') is the triangle formed by the intersection of the
line a = a' , p = p' and a = p . Differentiating Eq.(13) yields
^(a',P') = --
aa'ap'
(14)
Thus, the force mapping identification technique can be applied
to determine H(a,p) consistent with the experimental data, and
then ^(a,p) is obtained by Eq.(14).
5. Steady-State Response Analysis
Hysteretic systems are strongly nonlinear. A study of the
steady- state oscillation is one of the classical problems of
1050
nonlinear systems. Usually, the dynamic behavior of a nonlinear
system is represented by its resonant frequency and frequency
response characteristics. In the following, an accurate frequency-
domain method accommodating multiple harmonics is developed
to analyze the periodically forced response of hysteretic systems
defined by mathematical hysteresis models.
Fig. 3 shows a single-degree-of- freedom hysteretic oscillator
with mass m, viscous damping coefficient c, and linear stiffness
ky subjected to an external excitation F(t), for which the governing
equation of motion is
m • u(t) + c • u(t) + k ' u[t) + r{t) = F(t) (15)
where the hysteretic restoring force r(t) is represented by the DM
model as Eq.(4). It is worth noting that for the kinetic equation
Eq.(15), the excitation is F(t) and the response is u(t); and for the
hysteretic constitutive law Eq.(4), u(t) is input and r{t) is output.
The causal relationship is different.
Fig. 3 Single -Degree-of-Freedom Hysteretic System
Due to the hereditary nature of the hysteresis model, it is
difficult to directly solve the kinetic equation Eq.(15) by iteration.
Here, Eq.(15) is only used to establish the relation between the
harmonic components of u(t) and r(t). Suppose that the system is
subjected to a general periodic excitation F[t) with known
harmonic components F={Fq F^ F2 ••• F2 . The multi¬
harmonic steady-state response can be expressed as
a ^ ^ .
u{t) = ~ + a j cos j(Dt + a j sin jcot (16)
2 j=i
1051
in which a={ao a2 ••• a^v ^2 unknown
vector containing the harmonic components of u(t). Introducing
Eq.(16) into Eq.(15) and using the Galerkin method provide
ro^F^-k-a^ (17a)
Tj = Fj -c-citj-a] [j = 1,2, ••• , N) (17b)
Tj =Fj +c-coj-a^. -(/c-m-coV^)-a* (j = 1, 2, ,N) (17c)
where r={ro q ••• r^}'^ is the harmonic vector of
the hysteretic restoring force r(t). Referring to the hysteretic
constitutive law, we define the determining equation as
D(t) = r{t) - g[u,r,sgn(u)] • u{t) (18)
When a is the solution of u(t), applying the Galerkin method
into Eq.(18) and considering Eq.(17) achieve
d{a) = 0 (19)
where the vector d{a)={dQ ^3 --dj^ d^ d^ "• is comprised
of the harmonic components of D(t) corresponding to a. An
efficient procedure to seek the solution of Eq.(19) is the
Levenberg-Marquardt algorithm with the iteration formula
where the Jacobian matrix J[a(^)] = dd(a) / da\a=a^^) ; 9^ is the
Levenberg-Marquardt parameter and I is identity matrix.
At each iteration, the function vector and Jacobian
matrix should be recalculated with updated values of
Here, a frequency/ time domain alternation scheme by FFT is
introduced to evaluate the values of d(a) and J[a) at d{a)
and dd[a)/da are known to be the Fourier expansion coefficients
of D(t) and dD(t)/ da respectively. For a given a(^) and known F, the
corresponding r[a(^)] is obtained from Eq.(17), and the inverse
FFT is implemented for and r[a(^)] to obtain all the time
domain discrete values of u{t] , u(t) , r(t) and f(t) over an integral
period. Then the time domain discrete values of the function
D{u,u,r,r,t) , corresponding to a=a(^), are evaluated from Eq.(18).
Making use of forward FFT to these time domain discrete values
of D{u,u,r,r,t) , the values of function vector d[a(^)] are obtained.
1052
Similarly, the partial differential dD{t) / da can be analytically
evaluated in the time domain. Forward FFT to the time domain
values of dD(t) / da at ct=a(^) gives rise to dd[ai^)]/da.
6. Case Study
Wire-cable vibration isolators are typical hysteretic damping
devices. Dynamic tests show that their hysteresis behaviors are
almost independent of the frequency in the tested frequency
range [4,15]. Experimental study and parametric modelling of a
wire-cable isolator have been carried out [5]. Fig. 4 shows the
experimental hysteresis loops in shear mode. It is seen that for
relatively small deformations, the isolator exhibits softening
hysteresis loops. When large displacements are imposed, the
stiffness of the loops becomes smoothly hard. This nature is
referred to as soft-hardening hysteresis. Based on the Bouc-Wen
model, a parametric identification was performed to model these
hysteresis loops, but the result is unsatisfactory. This is due to
the fact that the Bouc-Wen model cannot represent such soft-
hardening nature of hysteresis.
Fig. 4 Experimental Hysteresis Loops
We now use the DM model to represent these hysteresis
loops, and perform a nonparametric identification to determine
the functions gi(u,7) and g2(u,7). The simplest Taylor series are
adopted, i.e. (i),(r) = (r / and (|)j(u) = (u j UqY~^ (ro = 20.0 and
uo = 2.0). Fig. 5 shows the identified “force” surfaces of gi{u,T) and
g2(u,r) by taking m=n~S. Fig. 6 presents the theoretical hysteresis
1053
loops generated by the DM model using the identified g\(u,T) and
g2(u,}). It is seen that the modeled hysteresis loops are agreeable
to the observed loops. In particular, the soft-hardening nature is
reflected in the modeled hysteresis loops.
-200
Fig. 6 Modeled Hysteresis Loops by DM Model
After performing the modelling of hysteretic behavior, the
dynamic responses of hysteretic systems can be predicted by the
developed method. Fig. 7 shows a vibration isolation system
installed with wire-cable isolators in shear mode. It is subjected
to harmonic ground acceleration excitation Xg[t)=Acos2Kft. The
equation of motion of the system is expressed as
m-u(t) + K ■r(t) = -rri' Xg(t) (21)
where m is the mass of the system; K is number of the isolators
installed. u{t) is the displacement of the system relative to the
ground. r(t) is the restoring force of each isolator and has been
determined from nonparametric identification.
Fig. 8 illustrates the predicted frequency-response curves of
the relative displacement when 7n=6kg and K=2. The excitation
amplitude A is taken as 0.25g, O.SOg, 0.35g, 0.40g and 0.45g
respectively. The frequency-response curves show clearly the
nonlinear nature of the wire-cable isolation system.
i
I
Xg(t)
Fig. 7 Vibration Isolation System with Wire -Cable Isolators
Fig. 8 Frequency Response Curves of Relative Displacement
1055
7. Concluding Remarks
This paper reports a preliminaiy work of introducing the
mathematical hysteresis models in structural and mechanical
areas. It is shown that a wide kind of differential hysteresis
models, which are extensively used at present, can be derived
from the Duhem-Madelung (DM) model. Thus, the mathematical
properties concerning the DM model are also possessed by these
models. Two potential advantages appear when the DM
formulation is used. Firstly, it allows to apply the force mapping
technique to hysteretic systems. Secondly, it provides an
approach to construct novel differential models which reflect
some special hysteretic characteristics. The Preisach model is
shown to be capable of representing nonlocal hysteresis and
mathematically tractable. It offers a more accurate description of
several observed hysteretic phenomena. Emphasis is placed on
demonstrating the selective-memory nature of this hysteresis
model. The case study based on experimental data of a wire-
cable isolator has shown the applicability of the mathematical
hysteresis model, and the validity of the steady-state response
analysis method proposed in the present paper.
Acknowledgment: This study was supported in part by the
Hong Kong Research Grants Council (RGC) and partly by The
Hong Kong Polytechnic University. These supports are gratefully
acknowledged.
References
1. Azevedo, J. and Calado, L., ‘‘Hysteretic behaviour of steel
members: analytical models and experimental tests”, J.
Construct Steel Research, 1994, 29, 71-94.
2. Kayvani, K. and Barzegar, F., “Hysteretic modelling of tubular
members and offshore platforms”, Eng. Struct, 1996, 18, 93-
101.
3. Lo, H.R., Hammond, J.K. and Sainsbury, M.G., “Nonlinear
system identification and modelling with application to an
isolator with hysteresis”, Proc. 6th Int modal Anal Conf.,
Kissimmee, Florida, 1988, Vol.II, 1453-1459.
1056
4. Demetriades, G.F., Constantinou, M.C. and Reinhorn, A.M.,
“Study of wire rope systems for seismic protection of
equipment in buildings”, Eng. Struct, 1993, 15, 321-334.
5. Ni, Y.Q., “Dynamic response and system identification of
nonlinear hysteretic systems”, PhD Dissertation, The Hong
Kong Polytechnic University, Hong, Kong, November 1996.
6. Krasnosefskii, M.A. and Pokrovskii, A.V., Systems with
Hysteresis, translated from the Russian by M. Niezgodka,
Springer-Verlag, Berlin, 1989.
7. Visintin, A., Differential Models of Hysteresis, Springer-Verlag,
Berlin, 1994,
8. Macki, J.W., Nistri, P. and Zecca, P., “Mathematical models
for hysteresis”, SIAM Review, 1993, 35, 94-123.
9. Masri, S.F. and Caughey, T.K., “A nonparametric
identification technique for nonlinear dynamic problems”, J.
Appl Mech, ASME, 1979, 46, 433-447.
10. Lo, H.R. and Hammond, J.K., “Nonlinear system identification
using the surface of nonlinearity form: discussion on
parameter estimation and some related problems”, Proc. 3rd
Int Conf Recent Adv. Struct. Dyn., Southampton, UK, 1988,
339-348.
11. Benedettini, F., Capecchi, D. and Vestroni, F., “Identification
of hysteretic oscillators under earthquake loading by
nonparametric models”, J. Eng. Mech., ASCE, 1995, 121, 606-
612.
12. Ni, Y.Q., Ko, J.M. and Wong, C.W., “Modelling and
identification of nonlinear hysteretic vibration isolators”.
Accepted to SPJE’s 4th Annual Symposium on Smart Structures
and Materials : Passive Damping and Isolation, 3-6 March
1997, San Diego, USA.
13. Ni, Y.Q., Wong, C.W. and Ko, J.M., “The generalized
orthogonal polynomial (GOP) method for the stability analysis
of periodic systems”, Proc. Int. Conf. Comput. Methods Struct.
Geotech. Eng., Hong Kong, 1994, Vol.II, 464-469.
14. Mayergoyz, I.D., Mathematical Models of Hysteresis, Springer-
Verlag, New York, 1991.
15. Ko, J.M., Ni, Y.Q. and Tian, Q.L., “Hysteretic behavior and
empirical modeling of a wire-cable vibration isolator”, Int. J.
Anal. Exp. Modal Anal, 1992, 7, 111-127.
1057
1058
The identification of turbogenerator foundation models
from run-down data
M Smart, M I Friswell, A W Lees, U Prells
Department of Mechanical Engineering
University of Wales Swansea, Swansea SA2 8PP UK
email: m.smart@swansea.ac.nk
ABSTRACT
The trend of placing turbines in modern power stations on flexi¬
ble steel foundations means that the foundations exert a considerable
influence on the dynamics of the system. In general, the complex¬
ity of the foundations means that models are not available a priori,
but rather need to be identified. One way of doing this is to use
the measured responses of the foundation at the bearings to the
synchronous excitation obtained when the rotor is run down. This
paper discusses the implementation of such an estimation technique,
based on an accurate model of the rotor and state of unbalance, and
some knowledge of the dynamics of the bearings. The effect of errors
in the bearing model and response measurements on the identified
parameters is considered, and the instrumental variable method is
suggested as one means of correcting them.
1 INTRODUCTION
The cost of failure of a typical turbine in a modern power station is
very high, and therefore development of condition monitoring techniques
for such machines is an active area of research. Condition monitoring relies
on measuring machine vibrations and using them to locate and quantify
faults, which obviously requires an accurate dynamic model of the ma¬
chine. Although the dynamic characteristics of rotors are generally well
understood, the foundations on which they rest are not. Since the founda¬
tions are often quite flexible, they can contribute considerably to the rotor’s
dynamic behaviour.
Finite element modelling has been attempted but the complexity of
the foundations, and the fact that they often differ substantially from the
original drawings rendered the technique generally unsuccessful [1]. Exper¬
imental modal analysis is another possible solution, but this requires that
the rotor be removed from the foundation, and that all casings remain in
place, which is not practical for existing power plant. However, mainte¬
nance procedures require that rotors are run down at regular intervals and
this procedure provides forcing to the foundation over the frequency range
1059
of operation. By measuring the response at the bearing pedestals (which
is already performed for condition monitoring purposes) an input-output
relation for the foundation may be obtained.
Lees [2] developed a least-squares method to calculate the foundation
parameters by assuming that an accurate model exists for the rotor, that
the state of unbalance is known from balancing runs, and that the dynamic
stiffness matrices of the bearings can be calculated. Although bearing mod¬
els are not in fact well characterised, Lees and Friswell[3] showed that the
parameter estimates are only sensitive to the bearing stiffnesses over limited
frequency ranges, which can be calculated.
Feng and Hahn [4] followed a similar approach but added extra informa¬
tion by measuring the displacements of the shaft. Zanetta[5] also measured
the shaft displacements but included the bearing characteristics as param¬
eters to be estimated. Although any extra information is desirable in a
parameter estimation routine the equipment necessary to measure these
quantities only exists in the newer power stations, and it was desired to
make the method applicable to older plant as well. In the analysis presented
here, the measured data consists of the motion of the bearing pedestals in
the horizontal and vertical directions, although not necessarily in both di¬
rections at every bearing.
2 THEORY
2.1 Force estimation
If D is the dynamic stiffness matrix of a structure defined as
D{u) K ^luC -oj^M (1)
where M,C,K are the mass, damping and stiffness matrices then
Dx = f (2)
where x is response and f force. Referring to figure 1 it is seen that the rotor
is connected to the foundation via the bearings. It is assumed that good
models exist for both rotor and bearings, and that the state of unbalance
is known. The implications of these assumptions will be discussed later.
The dynamic stiffness equation for the whole system may be written as
The subscripts r and / refer to the rotor and foundation degrees-of-freedom
respectively, u refers to the unbalance forces and b to the bearing forces.
1060
ROTOR
Figure 1: Rotor-bearing system
There is a negative sign before the bearing forces ft, since they refer to the
forces acting on the bearings. The foundation d.o.f are those where the re¬
sponses are measured, in other words no internal d.o.f are represented. Df
therefore represents a reduced dynamic stiffness matrix. The response mea¬
surements will not be the total vibration level at the bearings but rather the
vibrations at once-per-revolution and it is assumed that no dynamic forces
at this frequency will be transmitted to the foundation via the substructure
onto which it is fixed.
The equation for the bearings in the global coordinate system is
/ Dbrr Dbrf W ^7- ^ ^ fbT\
V Dij, Dill )\^s) \fbs)
(4)
This assumes that the bearings behave as complex springs, in other
words they have negligible inertia and no internal d.o.f. Substituting (4)
into (3) we have
/ Dt„ Dtrl \ ( ^'■ \ = (
V A/r A// J [ J A ° /
where
/ Dtrl 0 \ f Dirr Di^l \\
[Dtlr DtllJ-[[ 0 DlJ + [Diir Dill))
(5)
(6)
1061
and where subscript t refers to the total model. Solving equation (5) for
Xr leads to
Xr — J^trr ifur ~~
and solving equation (4) for ft / yields
fbf — ^fhr ~ ~^brr^r ~ ^brf^f (S)
All quantities in equation (8) are known either from assumed models {Dr, D^)
or experiment {xf, /^r)- This calculated force fbf may then be used to¬
gether with the measured responses to estimate the foundation parameters.
2.2 Foundation parameter estimation
Once the forces have been estimated, the foundation parameters must
be derived. The dynamic stiffness equation for the foundation is
DfXf — fbf (9)
Although D/ is a reduced stiffness matrix it is assumed that it has the
form of equation (1). Therefore equation (9) may be written as
W{u)v = fbf (10)
where is a column vector formed from the elements of M, C and K and
W is a matrix formed from the response vector which depends explicitly on
u. Clearly this is an under-determined set of equations, but by taking mea¬
surements at many frequencies it may be made over-determined, and thus
solvable in a least squares sense. Since the magnitude of the mass, damp¬
ing and stiffness elements normally differ by several orders-of-magnitude, it
was found expedient to scale the mass parameters by o;^, and the damping
parameters by uJ, where u is the mean value of the frequency.
2.3 Errors in estimates
It is necessary now to examine the effect of errors on the parameter esti¬
mates. Equation (10) is of the form Ax = b, where A has dimension mxn.
In this particular case, A depends on the measured response Xf, whilst b
depends on the measured response, applied unbalance, and assumed rotor
and bearing models. Therefore the estimated parameters will be sensitive
to the following errors:
1. Errors in the rotor model
2. Errors in the bearing model
3. Errors in the state of unbalance
1062
4. Errors in the measured foundation response
The rotor model is generally well known, as is the state of unbalance,
so the main source of error in the estimates is due to measurement noise
and bearing uncertainty. If the least squares problem is formulated as
Ax = b (11)
then
(Aq + Ajv)ic = bo + b/i/ (12)
where the subscript N refers to noise and 0 to data which is noise-free. The
least-squares estimate is given by
X = (Aq Ao + A^Aiv + + A^Ao) ^(Aq + Ayv)^(bo + b;v) (13)
Even if the noise on the outputs is uncorrelated with the noise on the inputs
the expected value of x does not equal that of its estimate:
E[x] - E[x] 0 (14)
In other words the estimate x is biased [6]. In order to reduce the bias of
the estimates, the instrumental variables method can be used. Essentially,
it requires the use of a matrix that is uncorrelated with the noise on the
outputs, but which is strongly correlated with the noise-free measurements
themselves. If W is the instrumental variable matrix, then
W'^Ax = W'^b (15)
Expanding
i = (W^Ao + + b^) (16)
This means that E[x] = x, in other words unbiased estimates result.
Fritzen[7] suggested an iterative method for solving for the parameters.
Initially, equation (10) is solved in a least-squares sense, and the values
of the estimated parameters are used to calculate outputs for the model.
These outputs are then used to create W in the same way as the original
outputs were used to create A, new estimates are obtained, and if neces¬
sary the process is repeated. Experience seems to suggest good convergence
properties[7].
3 SIMULATION
The method under discussion in this paper was tested on a model of a
small test rig located at Aston University, Birmingham. This consists of
a steel shaft approximately 1.1m long, with nominal diameter 38mm. The
shaft is supported at either end by a journal bearing of diameter 100mm,
1063
1 2 3 4 5 6 7 8 9 10 11 12
Figure 2: Rotor-bearing system
L/D ratio of 0.3 and clearance of 25/.tm. There are two shrink-fitted bal¬
ancing discs for balancing runs. Each bearing is supported on a flexible
pedestal to simulate the flexible foundations encountered in power station
turbines. At present these pedestals are bolted onto a massive lathe bed.
The rotor is powered by a DC motor attached via a belt to a driving pulley,
which is in turn attached via a flexible coupling to the main rotor shaft.
A schematic of the rig is shown in figure 2. Dimensions of each station
and material properties are given in table 1. A finite element model of the
rotor with 23 elements was created and short bearing theory was used to
obtain values for the bearing stiffness and damping[8].
The pedestals themselves consist of two rectangular steel plates, 600mm
X 150mm which have two channels cut into them, and which are supported
on knife-edges (figure 3). The vertical stiffness arises from the hinge effect
of the channels, whilst the horizontal stiffness is as a result of the shaft
centre tilting under an applied load. Treating the supports as beams, the
theoretical stiffnesses are:
Ky = 0.^5MN/m !<:, = 1.5MiV/m
where x and y refer to the horizontal and vertical directions respectively.
The masses and damping factors were taken as:
il/4 = My = 50/cp Ca: = Cy = 150iV • s/m
The estimation theory was tested using this model. The finite element
model was used to generate responses at the bearings for frequencies from
1064
Table 1: Table of rotor rig properties
Shaft Properties
Station
Length (mm)
Diameter (mm)
E (GPa)
P (kg/m^)
1
6.35
38.1
200
7850
2
25.4
77.57
200
7850
3
50.8
38.1
200
7850
4
203.2
100
200
7850
5
177.8
38.1
200
7850
6
50.8
116.8
200
7850
7
76.2
38.1
200
7850
8
76.2
109.7
200
7850
9
76.2
38.1
200
7850
10
50.8
102.9
200
7850
11
177.8
38.1
200
7850
12
203.2
100
200
7850
Balancing discs
Station
Length (mm)
Diameter (mm) Unbalance (kg • m)
6
25.4
203.2
0.001
25.4
203.2
0.001
0 to 30 Hz with a spacing of 0.1 Hz. The responses were corrupted by
normally distributed random noise with zero mean and standard deviation
of 0.1% of the maximum response amplitude (applied to both real and
imaginary parts of the response). At each frequency the bearing static
forces were disturbed by noise drawn from a uniform distribution spanning
an interval of 20% of the force magnitude, to introduce uncertainty into the
bearing parameters. The unbalance was assumed to be exactly known. A
series of 30 runs was performed, foundation parameter estimates calculated
and the mean and standard deviations of these estimates obtained.
The magnitudes of the responses at both bearings are given in figure 4,
which show that there are four critical speeds in the frequency range under
consideration. A sample of true and estimated forces in the bearings are
shown in figures 5 and 6.
The means {fi) and standard deviations (a) of the least-square (LS)
and instrumental variable (IV) estimates for the foundation parameters
are shown in table 2.
Displacement (m)
Figure 3: Flexible bearing supports
Figure 4: Magnitudes of responses at bearings
1066
Table 2; Parameter estimates for foundations showing uncertainty
1069
4 DISCUSSION
The results in table 2 show a clear improvement in parameter estimates
when the instrumental variable method is used. There is a clear bias in
the least-squares estimates which is significantly less when the instrumen¬
tal variable method is employed. Also, despite the fact that the bearing
parameters are assumed to be seriously in error, the estimates appear to be
insensitive to them. This will be true provided that the bearings are much
stiffer than the foundation (a reasonable assumption in practice). It does
appear however that in some cases the standard deviation of the instru¬
mental variable estimate is larger than that of the least-squares estimate,
a fact which warrants further investigation.
As far as the rotor model is concerned, impact tests, which are per¬
formed on rotors prior to them entering service, normally give experimental
frequencies which are within a few percent of the theoretical ones. Thus
the assumption that the rotor model is accurately known would appear to
be reasonable.
The state of unbalance may in theory be established from a balancing
run. If two successive run-downs are performed, one due to the unknown
system unbalance and one with known balance weights attached, then pro¬
vided the system is linear the response measurements may be vectorially
subtracted to give the response due to the known balance weights alone.
In order to ascertain the effect of unbalance uncertainty on the parameter
estimates, one run was performed assuming no error in the unbalance. It
should be noted that this assumes that the system is linear.
5 CONCLUSIONS
A method of estimating turbogenerator foundation parameters from
potentially noisy measurement data is demonstrated. It is shown that
making use of the instrumental variable method reduces the bias in the
estimates and improves them quite significantly.
6 ACKNOWLEDGEMENT
The authors wish to acknowledge the support and funding of Nuclear
Electric Ltd and Magnox Electric Pic. Dr Friswell wishes to acknowledge
the support of the Engineering and Physical Sciences Research Council
through the award of an advanced fellowship.
1070
REFERENCES
[1] A. W. Lees and I. C. Simpson. The dynamics of turbo-alternator foun¬
dations: Paper C6/83. In Conference on steam and gas turbine founda¬
tions and shaft alignment, Bury St Edmunds, 1983, IMechE, pp37-44.
[2] A. W. Lees. The least squares method applied to identify ro¬
tor/foundation parameters: Paper C306/88. In Proceedings of the Inter¬
national Conference on Vibrations in Rotating Machinery, Edinburgh,
1988, IMechE, pp209-216.
[3] M. 1. Friswell and A. W. Lees. Estimation of forces exerted on machine
foundations. In M. 1. Friswell and J. E. Mottershead, editors, Interna¬
tional Conference on Identification in Engineering Systems, Swansea,
1996, pp793-803.
[4] N. S. Feng and E. J. Hahn. Including foundation effects on the vibra¬
tion behaviour of rotating machinery. Mechanical Systems and Signal
Processing, 1995, 9, pp243-256.
[5] G. A. Zanetta. Identification methods in the dynamics of turbogener¬
ator rotors: Paper C432/092. In IMechE Conference on Vibrations in
rotating machinery, Bath, 1992. IMechE, ppl73-181.
[6] J. Schoukens and R. Pintelon. Identification of linear systems. Perga-
mon Press, 1991.
[7] C. P. Fritzen. Identification of mass, damping and stiffness matrices of
mechanical systems. Journal of Vibration, Acoustics, Stress and Relia¬
bility if Design, 1986 108, pp9-17.
[8] D. M. Smith. Journal bearings in Turbomachinery. Chapman and Hail,
1969.
1071
1072
SHELL MODE NOISE IN RECIPROCATING REFRIGERATION
COMPRESSORS
Ciineyt Oztiirk and Aydin Bahadir
Tiirk Elektrik Endiistrisi A.$
R&D Department
Davutpa§a, Litres Yolu, Topkapi -34020, Istanbul, Turkey
ABSTRACT
This study describes the successful endeavor to understand the causes of noise
that appear on the shell modes of the reciprocating refrigeration compressors.
The compressor shell is generally considered as the acoustic enclosure that
reflects the acoustic energy back into the compressor cavity but also as the
transmitter and radiator of the transmitted acoustic energy that could be
radiated into the air or transmitted to the structure. Vibrations of the
compressor shell can easily be characterized in terms of the modal parameters
that consist of the natural frequencies, mode shapes and damping coefficients.
The noise source harmonics and the shell resonances couple to produce the
shell noise and vibration. The harmonic spacing is equal to the basic pumping
frequency. Results of the studies indicate that important natural frequencies of
the compressor shell usually stay between 2000-6000 Hz interval. The
important natural frequencies are first natural frequencies in the lower range
with the longer wavelengths that radiate well.
INTRODUCTION
Compressor noise sources are those processes where certain portions are
separated from the desired energy flow and transmitted through the internal
components of compressor to the hermetic shell where it is radiated from the
shell as airborne noise on vibration of supporting structure will eventually
radiate noise from some portion of the structure. Noise sources of the
reciprocating refrigeration compressors can be classified as motor noise,
compression process noise and valve port flow noise.
1073
In reciprocating compressors there is very high density of noise harmonics even
though they decay in amplitude at high frequencies. Generally, these noise
source harmonics and the shell resonances couple to produce shell noise and
vibration
NOISE GENERATION MECHANISM OF THE RECIPROCATING
REFRIGERATION COMPRESSORS
Significance of the problem
The results of the sound radiation characteristics shown at figure 1 indicated
that certain high amplitude frequency components had very distinctive sound
radiation patterns. It was suspected these frequencies correspond to excitation
of either structural resonances of the compressor shell or acoustic resonances of
the interior cavity space. Resonances those amplify the noise and vibration
caused by pumping harmonics of a compressor and thus can be the cause of
significant noise problems.
Sound power - A-Wcighted
— I - , - 1 - , - ; - P — — r — f — ! — — 'I' "I '"'I — I - ”r ' i i
50.0 63.0 80,0 100,0 125,0 160,0 200,0 250,0 315,0 400,0 500,0 630,0 800,0 l,0k t,3k 1,6k 2,0k 2,5k 4,0k S,0k 63k S,0k 10,0k
Frequency [Hz]
Figure 1, Noise Radiation Characteristics of Reciprocating Refrigeration
Compressor.
Noise sources
Noise in a compressor is generated during cyclic compression, discharge,
expansion and suction process. The character of noise sources is harmonic due
to periodic nature of the compression process. These harmonics are present in
the compression chamber, pressure time history and loading of the compressor
through drive system. The motor can not provide immediate response to load
harmonics and load balance is obtained at the expense of acceleration and
1074
deceleration of the motor drive system. Harmonic vibrations of the motor drive
system can then excite the resonant response in the compressor components
that can transmit the acoustic energy in very efficient way. The rest of the noise
sources are, turbulent nature of flow depending to passage through valve ports,
valve impacts on their seats and possible amplification when matched with
mechanical resonances.
MECHANICAL FEATURES OF SHELL
Mechanical features of the compressor
The hermetically sealed motor compressor comprises in general a motor
compressor unit including a motor assembly mounted with a frame and a sealed
housing within which the compressor is supported by means of plurality of coil
springs each having one end spring with the frame and the other end connected
with the interior of the housing.
The refrigerant gas as it is compressed in the cylinder is discharged through the
discharge chamber in the cylinder head into the discharge muffler. The
discharge muffler is generally mounted on the cylinder head attached in
covering relation to an end face of the cylinder. Where the sealed casing is
spherical in shape for better noise suppression, an upper end of the cylinder
head tends to interfere with an inner wall surface of the casing, a disadvantage
that can only be eliminated by increasing the size of scaled casing for providing
a desired hermetically sealed motor.
Compressor Shell
The shell is easily be characterized with the modal parameters. The ideal shell
should be designed in a way that keep all the excitation frequencies at the mass
controlled region of all its modes. But, depending on the very tight constraints
that come with the gas dynamics and motor locations, it is not allowed to be
flexible during the design of shell. As a consequence of the existing design
limitations natural frequencies of the shell usually fall between 1000-5000 Hz.
SHELL RADIATION
Figure 1 illustrates how the sound pressure level of a pumping harmonic can
increase as it nears a resonant frequency. The sound pressure level of the
pumping harmonic increases around the shell resonances. The resonance
generally radiates primarily from the large flat sides of the compressor. There
are three major acoustic cavity resonances 400, 500 and 630 Hz and four major
structural resonances of the compressor shell: around 1.6 K, 2 K, 2.5 and 3.2
K. Hz at which noise radiates well in certain directions.
1075
SHELL RESONANCES
Figures 2, 3 and 4 illustrate the frequency responses of the compressor shell
when measured with the impact hammer method. Figure 2 is the response to
the excitation in x direction, figure 3 is for the excitation direction and figure 4
is for the excitation in z direction.
Figure 2, Frequency response of the compressor shell when excited in X
direction.
Figure 3, Frequency response of the compressor shell when excited in Y
direction.
1076
Figure 4, Frequency response of the compressor shell when excited in Z
direction.
To verify the hypothesis that resonances were contributing to some of the noise
problems of the reciprocating piston compressor, a modal analysis of the shell
and interior cavity was performed.
For the shell modal analysis, the accelerometer to measure the response
remained stationary, while the impact location was moved. The test was
performed in this manner for convenience since it was easier to fix the
accelerometer in one location and strike the compressor with force hammer at
each grid point to obtain transfer function for each measurement location.
Identical results are obtained if impact occurred at a single point and the
response was measured at each measurement location. Preliminary test were
performed initially to identify an appropriate measurement location at which all
important natural frequencies of shell are detectable. Several force input and
response locations were evaluated to determine the best location to mount
accelerometer to measure the shell response.
The shell resonances are also calculated by using the Structural FEM analysis.
Table- 1 lists the natural frequencies predicted in these studies. During the finite
element analysis, the models of the compressor were built, based on the CAD
models. The shell consists of 7500 elements. The mesh densities are quite
adequate for the structural analysis in the frequency range of interest. In order
to investigate the possible influence of the crank mechanism on the natural
frequencies of the shell, a simple model of the crank mechanism was introduced
to the FE model. During the calculations, the crank mechanism was simplified
as a rigid block with certain mass and rotary inertia and modeled with solid
elements. The shell and the crank mechanism have been suspended with the
1077
springs from 3 positions and in all 6 transitional and rotational directions.
Depending on the negligible spring effect on the longitudinal direction,
estimated values have been used in 5 directions. The FE model has been
assumed to be free-free.
Mode
#
Frequency
Hz
Mode
#
Frequency
Hz
1
1997
21
4716
2
22
4925
3
2293
23
4994
4
24
5
25
5159
6
26
5454
7
2889
27
5476
8
3258
28
9
29
EESI^HI
10
30
5783
11
3376
31
5936
12
3551
32
5999
13
3577
33
6035
14
34
6055
15
3788
35
6183
16
3958
36
6237
17
4383
37
6314
18
4481
38
19
4644
39
20
4702
40
6701 1
Table- 1, Calculated Natural Frequencies of the Shell
Figures 5, 6, 7 and 8 illustrate how the mode shapes of the shell vary at the
mode frequencies of 2754, 3332, 3551 and 3788 Hz , These figures indicate
that the shell vibrates predominantly along the large flat sides of the compressor
at points where the suspension springs are attached to the shell wall at these
natural frequencies. When referred to figure 1 of the noise radiation this
frequency range is also the range where the noise radiates efficiently from the
large flat sides of the compressor. Thus, there is good correlation between the
acoustic data and structural data for these frequencies. The slight discrepancies
in the structural natural frequencies and the acoustic data. Acoustic data have
been picked up at the shell temperature of the reciprocating piston compressor
that could reach up to 1 10 C.
The modal analysis results also indicate that the compressor suspension springs
are attached to a point on the shell where the shell is comparatively compliant.
Thus, the vibrational energy transmitted through the springs to the compressor
shell can and did effectively excite the shell vibrations. Also, significant shell
vibrations occur along the large flat sides of the compressor shell indicating the
curvature of the shell needs to be increased to add stifihiess to the shell.
Based on the results of the shell modal analysis, it is recommended the
suspension springs moved away from the compliant side walls of the shell. A
four spring arrangement at the bottom of the shell near corners where the
curvature is sharp would reduce the amount of vibration energy transferred to
the shell because of the reduced input mobility of the shell at these locations.
It is also believed increasing the stiffness of the shell by increasing the curvature
will provide noise reduction benefits. The greater shell stiffness lowers the
amplitude of the shell vibrations. Figure 9, illustrate the third octave change in
compressor noise with the same compressor in the new shell. An over all noise
level of 5 dBA has been obtained.
Figure 9 Compressor noise level improvement after the shell modification.
The increased shell stiffness also raises the natural frequencies of the shell
where there is less energy for transfer function response. However, there is a
possible disadvantage to increasing stiffness of the shell. The higher natural
frequency lowers critical frequency of the shell thus reducing transmission loss
of the shell.
1081
Damping treatments can also have obvious benefits in vibration and noise
reduction. Visco elastic and Acoustic dampings are considered to avoid the
shell excitations. The application of dampers can also provide up to 5 dBA
reductions when appropriately located on the shell.
CONCLUSION
The results of studies indicate that structural resonances of the shell are indeed
amplifying the noise due to the pumping harmonics of the reciprocating
refrigeration compressor to cause significant noise radiation outside of
compressor.
In order to tackle with this noise problem, within the scope of these studies
two different effective shell noise control are considered based on the results of
numerical and experimental structural analysis and acoustic features of the
reciprocating compressor. First, shell noise control method is the redesign of
the shell with increased stiffness by replacing all the abrupt changes in the
curvature with the smooth continuous changes. It is apparent that change in the
shell configuration can shift the first shell resonance from 1750 Hz to nearly
3200 Hz. The results of the redesign effects can reach up to 3-5 dBA reduction
on third octave noise levels. Second treatment that could be applied against the
excitation of shell resonances are considered as the acoustic and viscoelastic
dampers. These dampers can be chosen to operate efficiently at the shell
resonant frequencies. These two applications can also provide up to 2-5 dBA
reduction on the third octave band of the noise emission but the long term
endurance and temperature dependence of these materials can always be a
question mark when considered from the manufacturer point of view.
REFERENCES
1. JFROBATTAand ID. JONES 1991, Purdue University, School of
Mechanical Engineering, The Ray Herrick Laboratories, Report no : 1912-1
HL 91-9P, 73-84, Investigation of Noise Generation Mechanisms and
Transmission Paths of Fractional Horsepower Reciprocating Piston and
Rolling Piston Compressors
2. HAMILTON I F 1988, Purdue University , School of Mechanical
Engineering, The Ray Herrick Laboratories, 207-213 Measurement and
Control of Compressor Noise
3. C OZTURK, A AQIKGOZ and J L MIGEOT 1996, International
Compressor Engineering Conference at Purdue, Conference Proeceeding ,
Volume II, 697-703, Radiation Analysis of the Reciprocating Refrigeration
Compressor Casing
1082
A COMPARATIVE STUDY OF MOVING FORCE IDENTIFICATION
T.H.T. Chan, S.S. Law, T.H. Yung
Department of Civil & Structural Engineering,
The Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong
ABSTRACT
Traditional ways to acquire truck axle and gross weight information are
expensive and subject to bias, and this has led to the development of Weigh-
in-Motion (WIM) techniques. Most of the existing WIM systems have been
developed to measure only the static axle loads. However dynamic axle loads
are also important. Some systems use instrumented vehicles to measure
dynamic axle loads, but are subject to bias. These all prompt the need to
develop a system to measure the dynamic axle loads using an unbiased
random sample of vehicles. This paper aims to introduce four methods in
determining such dynamic forces from bridge responses. The four methods are
compared with one another based on maximum number of forces to be
identified, minimum number of sensors, sensitivity towards noise and the
computation time. It is concluded that acceptable estimates could be obtained
by all the four methods. Further work includes merging the four methods into
a Moving Force Identification System (MFIS).
INTRODUCTION
The truck axle and gross weight information have application in areas
such as the structural and maintenance requirements of bridges and pavements.
However, the traditional ways to acquire that are expensive and subject to bias,
and this has led to the development of Weigh-in-Motion (WIM) techniques.
Some systems are road-surface systems which make use of piezo-electric
(pressure electricity) or capacitive properties to develop a plastic mat or
capacitive sensors to measure axle weight [1]. Another kind of WIM system is
the under-structure systems in which sensors are installed under a bridge or a
culvert and the axle loads are computed from the measured responses e.g.
AXWAY [2] and CULWAY [3]. All the above mentioned systems can only
give the equivalent static axle loads. However dynamic axle loads are also
important as they may increase road surface damage by a factor of 2 to 4 over
that caused by static loads [4]. Some systems use instrumented vehicles to
measure dynamic axle loads [5], but are subject to bias. These all prompt the
need to develop a system to measure the dynamic axle loads using unbiased
random samples of vehicles. Four methods are developed to determine such
1083
dynamic forces from bridge responses which include bending moments or
accelerations.
EQUATION OF MOTIONS FOR MOVING LOADS
The moving force identification methods described in this paper are the
inverse problems of an predictive analysis which is defined by 0‘ Connor and
Chan [6] as an analysis to simulate the structural response caused by a set of
time-varying forces running across a bridge. Two models can be used for this
kind of analysis.
A. Beam-Elements Model
0‘ Connor and Chan [6] model the bridge as an assembly of lumped
masses interconnected by massless elastic beam elements as shown in Figure 1,
and the nodal responses for displacement or bending moments at any instant
are given by Equations (1) and (2) respectively.
Moving Loads
... ip. Ip, ^
- D — D — 2^
Lumped Masses 1 2 ... N-1 N
Figure 1 - Beam-Elements Model
{Y} = [Y,]{F}-lY,][Am]{n-m[C]{Y} (1)
{5M} = [5MJ{P}-[5M;][Am]{y}-[5MJ[C]{f} (2)
where [P] is a vector of wheel loads, [Am] is a diagonal matrix containing
values of lumped mass, [C] is the damping matrix, 5M,L,7,7are the nodal
bending moments, displacements, velocities and accelerations respectively,
[R,^] {'R ’ can be Y or BAP) is an m x n matrix with the ith column representing
the nodal responses caused by a unit load acting at the position of the ith wheel
load and [i?J {'R' can be Y or BM) is an n x n matrix with the ith column
representing the nodal responses caused by a unit load acting at the position of
ith internal node.
1084
B. Continuous Beam Model
Assuming the beam is of constant cross-section with constant mass per
unit length, having linear, viscous proportional damping and with small
deflections, neglecting the effects of shear deformation and rotary inertia
(Bernoulli-Euler’s beam), and the force is moving from left to right at a
constant speed c, as shown in Figure 2, then the equation of motion can be
written as
P
d-v{xj)
a-
a
+ EI
^v(x,0
a-
S(x-ct)f{t)
(3)
where v(x,0 is the beam deflection at point x and time t; p is mass per unit
length; C is viscous damping parameter; £ is Young's modulus of material; I
is the second moment of inertia of the beam cross-section; f(t) is the time-
varying force moving at a constant speed of c, and Sft) is Dirac delta function.
\/(0
— O
Figure 2 - Simply supported beam subjected to a moving force f(t)
Based on modal superposition, the dynamic deflection y(x,t) can be
described as follows;
v{x,t) = Y,<i'„{x)q„{t)
}}=\
(4)
where n is the mode number; 0n(x^) is the mode shape function of the n-th
mode and qyi(t) is the n-th modal amplitudes.
Based on the above mentioned predictive analyses, four Moving Force
Identification Methods (MFIM) are developed.
1085
FIRST MOVING FORCE IDENTIFICATION METHOD
INTERPRETIVE METHOD I - BEAM-ELEMENTS MODEL (IMI)
It is an inverse problem of the predictive analysis using beam-elements
model From Equation (1), it can be seen that if Y is known at all times for all
interior nodes, then 7 and Y can be obtained by numerical differentiation.
Equation (1) becomes an overdetermined set of linear simultaneous equations
in which the P may be solved for them. However a particular difficulty arises
if measured BM are used as input data. Remembering that the moving loads P
are not normally at the nodes, the relation between nodal displacements and
nodal bending moments is
{r}^[Y,]{BM} + [Yc]{B} (5)
where allows for the deflections due to the additional triangular
bending moment diagrams that occur within elements carrying one or more
point loads P. [}^]can be calculated from the known locations of the loads.
[}^^] and {BM}aic known, but {7} cannot be determined without a
knowledge of (P}. 0‘Connor and Chan [6] describe a solution uses values of
{P} assumed from the previous time steps.
SECOND MOVING FORCE IDENTIFICATION METHOD
INTERPRETIVE METHOD II - CONTINUOUS BEAM MODEL (IMII)
From the predictive analysis using continuous beam model, if the ith-
iTTX
mode shape function of the simply supported Euler‘s beam is sin-—-, then
the solution of Equation (3) takes the form
v = ^sin— F;(0 (6)
/=i L
where V.(t) , (i = 1, 2, • • •) are the modal displacements.
Substitute Equation (6) into Equation (3), and multiply each term of
Equation (3) by the mode shape function sin(y;r.x / L) , and then integrate the
resultant equation with respect to x between 0 and L and use the boundary
conditions and the properties of Dirac function. Consequently, the following
equation can be obtained
1086
op
u 2 El ^
where = —5 — - C(y, =
C
2/rryy,
at the j-th mode.
If there are more than one moving loads on the beam, Equation (7) can be
written as
. 7r(ct-Xk)
._7r(ct~x,)
7r{ct-x^) ■
■^.1
'2CM'
'cojv;
sin - ^
L
sin - ^ •
L
sin - —
L
'K
V,
2C,co,K
+
CO IVj
_ 2
. 27r(ct-Xt)
sin
. 27r(ct~x^}
sm - =- ■
. 27r{ct - x. )
•• sm — - —
Pi
~ Ml
A
.1
_2C,.a)/„_
_coX_
. n7r(ct-x,)
sin — ^ ^
. riKict-x^)
sin — ^ - — .
. n7c{ct-x^)
•• sin — - —
A.
L
L
L
(8)
in which is the distance between the k-th load and the first load and x, = 0 .
If Pi Pk are constants, the closed form solution of Equation (3) is
/ N ^ 1 • ■ j7E{ct-Xi) a . . . , ^ {Q\
48E/f:i' L \ L j ^ J
nc
in which a =
Lcd
If we know the displacements of the beam at x, , x, , • • • , , the moving
loads on the beam are given by
{v} = [S, „.]{/>} (10)
in which {v}=[v| v, ■■■ v,]' {•^’} = [-^1 ^2 ■"
^n-
, where
L" A 1 . j7rx,J . J7r{ct-x.) a . , . , ,
= 7577X^2772 - 27^“ “7~ - 7 - /c)
48£:/^;-0- -a ) L \ L j
1087
If l>k, that means the number of nodal displacements is larger than or
equal to the number of axle loads, then according to the least squares method,
the equivalent static axle load can be given by
{/>}=([s,,]''[s„])''[s,q^{v} (11)
If the loads are not constant with time, then central difference is used to
proceed from modal displacements to modal velocities and accelerations.
Equation (8) becomes a set of linear equations in which P^. for any instant can
be solved by least squares method. Similar sets of equations could be obtained
for using bending moments to identify the moving loads.
THIRD MOVING FORCE IDENTIFICATION METHOD
TIME DOMAIN METHOD (TDM)
This method is based on the system identification theory [7]. Substituting
Equation (4) into Equation (3), and multiplying each term by 0j(x), integrating
with respect to x between 0 and L, and applying the orthogonality conditions,
then
dt-
dt
M.,
(12)
where con is the modal frequency of the n-th mode; is the damping ratio of
the n-th mode; Mn is the modal mass of the n-th modt, pn(t) is the modal force
and the mode shape function can be assumed as 0„(x) = sin(^;w / L) .
Equation (12) can be solved in the time domain by the convolution
integral, and yields
= (13)
^11 0
where ~ sin(^u„0, t >0 (14)
and co\, (15)
Substituting Equation (13) into Equation (4), the dynamic deflection of
the beam at point x and time t can be found as
v(x,0 = y — ^sin^^ sin6>|,(^ - '^)sin^^^^/(T)dr (16)
fxpLo),, L i L
1088
A. Force Identification from Bending Moments
The bending moment of the beam at point x and time t is
m{xj) = -El -
3c-
(17)
Substituting Equation (16) into Equation (17), and assuming the force f(t)
is a step function in a small time interval and f(t) =0 at the entry and exit, then
let
^ 2El7r' n~ . yitu: ^
C,„ = — 73 - ^sm— Ar,
pi co„ L
{k) = sin(<:y'„ A/A:),
^ . ,n7(cNi
52(*) = sm(— ^/c)
(18)
(19)
N„=-
Equation (17) can be expressed as
cEt
m(2)]
0
... 0 '
/(i)
m(3)
<
■=S c,„
/) = 1
E;,S,{2)S,i\)
e:S,{\)S,{2)
... 0
/(2)
m{N)
_E::"S,iN-\)S,il)
E;^--S,{N~2)S,i2)
Kc.
(20)
where A^ is the sample interval and N+I is the number of sample points, and
f>„=£r'''“*'S,(Al-A^„+l)S,(//„-l)
Equation (20) can be simplified as
B f = m (21)
(A'-l)x(;V/^-|) ('V/^->)xl (,V_l)xl
U N = matrix B is a lower triangular matrix. We can directly find
the force vector f by solving Equation (21). If 77 > and/or 77/ bending
moments (77/ > 1) are measured, least squares method can be used to find
the force vector f from
1089
(22)
B, ■
m,
f
m.
®-v,.
The above procedure is derived for single force identification. Equation
(21) can be modified for two-forces identification using the linear
superposition principle as
“B. 0
B, B,
B. B,
where B 3 [N^x (Nb-1)] , Bt [(N -1-2NJx (Ng-l)], and B^ [NjX (Nb -1)] are
sub-matrices of matrix B. The first row of sub-matrices in the first matrix
describes the state having the first force on beam after its entry. The second
and third rows of sub-matrices describe the states having two-forces on beam
and one force on beam after the exit of the first force.
B. Identification from Bending Moments and Accelerations
Similarly the acceleration response of the beam can be expressed as
A f = V
(24)
The force can also be found from the measured acceleration from
Equation (24). If the bending moments and accelerations responses are
measured at the same time, both of them can be used together to identify the
moving force. The vector m in Equation (21) and v in Equation (24) should be
scaled to have dimensionless unit, and the two equations are then combined
together to give
A/iv||J [v/llvlj
(25)
where Ihli is the norm of the vector.
FOURTH MOVING FORCE IDENTIFICATION METHOD
FREQUENCY DOMAIN METHOD (FDM)
Equation (12) can also be solved in the Frequency Domain. Performing
the Fourier Transform for Equation (12),
1090
1
1
(26)
where
co;, -CO- +24„co„co M„
W
i
—00
(27)
(28)
Let
_ 1 _
co;,-co- +2^„a„a
(29)
Hn(co) is the frequency response function of the n-th mode. Performing the
Fourier Transform of Equation (4), and substituting Equations (26) and (29)
into the resultant equation, the Fourier Transform of the dynamic deflection
v(x,t) is obtained as
A. Force Identification from Accelerations
Based on Equation (30), the Fourier Transform of the acceleration of the
beam at point .x and time t can be written as
V(x,a) = -®^X^(D„(x)//„(cy)P„(cy) (31)
Considering the periodic property of the Discrete Fourier Transform (DFT),
and let
A/^m-
Equation (32) can be rewritten as
(32)
1091
,m = 0,l, (33)
V{m) = X H„(m)'i'„(m)[F,{0) + iF,(0)]
/;=1
A^/2-1 « _
k=\ H=t
Nn~\ =0 _
/t = l /) = l
+E /2)[F„(A^ /2) - iivCA^ / 2)]
«=I
where is the Fourier Transform of the n-th mode shape, and F is the
Fourier Transform of the moving force.
Writing Equation (33) into matrix form and dividing F and V into real and
imaginary parts, it yields
Because F, (0) = 0, Fj (N / 2) = 0,Vj (0) = 0,F; ( / 2) = 0 , Equation (34) can
be condensed into a set of N order simultaneously equations as
(35)
Fr and F, can be found from Equation (35) by solving the Nth order linear
equation. The time history of the moving force f(t) can then be obtained by
performing the inverse Fourier Transformation.
If the DFTs are expressed in matrix form, the Fourier Transform of the
force vector f can be written as follows if the terms in f are real [8].
F = — Wf (36)
N
whereW = e''^‘‘^^^ (37)
1092
0 0 0 0 0
0 1 2 ••• -2 -1
0 2 4 ••• -4 -2
0 -2 -4 ••• 4 2
0 -1 -2 2 1
yv X
The matrix W is an unitary matrix, which means
w-‘ =(w* y
(38)
where W* is a conjugate of W. Substituting Equation (36) into Equation (35),
V = — Al
N
W 0
- 1
X
_ Ij
(39)
or
V=^ A fe
(40)
linking the Fourier Transform of acceleration V with the force vector fg of
the moving forces in the time domain. Wg is the sub-matrix of W. If N =
fg can be found by solving the Mh order linear equations. If N> or more
than one accelerations are measured, the least squares method can be used to
find the time history of the moving force f(t).
Equation (40) can be rewritten as follows
(41)
relating the accelerations and force vectors in the time domain. Also if N -
Nb, fe can be found by solving the Nth order linear equation. If N > Nb or
more than one acceleration are measured, the least squares method can be
used to find the time history of the moving force f(t).
If only Nc {Nc ^ N) response data points of the beam are used, the
equations for these data points in Equation (41) are extracted, and described as
V, =(w*)" A
C V B / A/x/V
c
B
Nr X N
W3
/Vx;V,
yV„x!
(42)
1093
In usual cases Nc > Ng, so the least squares method is used to find the time
history of the moving force f(t). More than one acceleration measurements at
different locations can be used together to identify a single moving force for
higher accuracy.
B. Identification from Bending Moments and Accelerations
Similarly, the relationships between bending moment m (and M ) and
the moving force f can be described as follows,
M = ^ B W is (43)
/Vxl N N^Nn
(44)
(45)
The force vector fg can be obtained from the above three sets of
equations. Furthermore, these equations can be combined with Equations (40),
(41) and (42) to construct overdetermined equations before the equations are
scaled. Two forces identification are developed using the similar procedure as
that for the Time Domain Method.
COMPARATIVE STUDY
The first moving force identification method is implemented in a
computer program using FORTRAN, while the other three methods are
implemented under the environment of a high performance numerical
computation and visualization software. The predictive analysis using beam-
elements model is used to generate the theoretical bridge responses and the
four moving force identification methods then use these responses to recover
the original dynamic loads. In this study, if at least 80% of the identified
forces at any instant of any load lie within ± 1 0% of the original input force,
the method is considered acceptable. It is found that all the four methods can
give acceptable results.
It is decided to carry out a preliminary comparative study on the four
methods in order to study the merits and limitations of each method so as to
consider the future development of each method and devise a plan to develop a
1094
moving force identification system which can make use of the benefits of all
the four methods.
A. Maximum Number of Forces
This is to examine the maximum number of axle loads that can be
identified by each method. Theoretically, provided that sufficient number of
nodal sensors are installed, IMI and IMII can be used to identify as many loads
as the system allows. Basically, the number of axle loads cannot be larger than
the number of nodal sensors. Regarding TDM and FDM, as the formulation of
the governing equation is derived for two moving forces, the maximum
number of axle loads that can be identified is two.
B. Minimum Number of Sensors
Based on a study of common axle spacings of vehicles currently
operating on Australian roads, and the cases with zero nodal responses,
0‘ Connor and Chan [6] state the relationships of the minimum number of
sensors used for IMI and the span length of a bridge as follows:
Using bending moment, for span length L > 4.8m,
Min. number of nodal moments required = int{
Using displacements, for span length L > 13.8m,
Min. number of nodal displacement required = INj{
and for span length L <13.8m,
Min. number of nodal displacement required
L - 4.8'
1.7
L-UX
3.7
+ 4
+ 6
(46)
(47)
(48)
For IMII, it is found that the number of sensors required are generally less
than that for IMI. Regarding TDM and FDM, the programs are not as flexible
as that for IMI and IMII and it is not easy to change the number of sensors.
Meanwhile the sensors are fixed to be at 1/4, 1/2 and 3/4 of the span.
C. Sensitivity towards Noise
In general, all the four methods can compute the identified forces exactly
the same as those given to the predictive analysis to generate the
corresponding responses. It is decided to add white noise to the calculated
responses to simulate polluted measurements and to check their sensitivity
towards noise. The polluted measurements are generated by the following
1095
equations:
m = n>o.nkui«.«] x N„,
(49)
where Ep is a specified error level; is a standard normal distribution
vector (with zero mean value and unity standard deviation).
Several cases are studied using Ep =1%, 3%, 5% and 10%. It is found
that when using bending moments for IMI and IMII, and if Ep is less than 3%,
acceptable results can be obtained. For noise which is greater than 3%, a
smoothing scheme should be adopted to smooth the simulated data.
Acceptable results cannot be obtained for Ep > 10%. Besides, both IMI and
IMII cannot give acceptable results when using displacements.
Both TDM and FDM cannot give acceptable results when using
displacements only, accelerations only or bending moments only. In general
TDM and FDM are less sensitive to noise when comparing to IMI and IMII.
They can give acceptable results for Ep up to 5 % without any smoothing of
the polluted simulated data.
D. Computation Time
In general, the computer program for IMI only takes few seconds to
identify moving forces. In order to compare the computation time, IMI is
implemented in the same environment as the other three methods. It is found
that IMI and IMII take about 2-3 minutes to give the identified forces for a
case of two axle loads using a 80486 computer. However, under the same
working conditions, TDM and FDM almost take a whole day for any one of
them to identify two moving forces. It is due to the fact that both of them
require to set up an huge parametric matrix.
CONCLUSIONS
Four methods are developed to identify moving time-varying force and
they all can produce acceptable results. From a preliminary comparative study
of the methods, it is found that IMI and IMII have a wider applicability as the
locations of sensors are not fixed and it can identify more than two moving
forces. However, TDM and FDM are less sensitive to noise and require less
number of sensors. It is decided to further improve the four methods and then
a more detailed and systematic comparison can be carried out afterwards. The
possible development of the methods are described as follows.
1096
Both the IMI and IMII are developed to work with one kind of responses,
e.g. either displacements or bending moments. It is suggested to modify the
programs to use mixed input parameter, e.g. use bending moments as well as
accelerations as that for TDM or FDM. Regarding the TDM and FDM, as the
basic formulations are based on two-axle moving forces, so it is necessary to
modify the governing equations for multi-axle. In addition, the computation
time for TDM or FDM under the environment of the high performance
numerical computation and visualization software used is unbearable. It is
expected that the time will be significantly reduced if the methods are
implemented in programs using standard programming languages like
FORTRAN 90 or C. Then the four methods can be combined together and
merged into a Moving Force Identification System (MFIS) so that it can
automatically select the best solution routines for the identification.
ACKNOWLEDGMENT
The present project is funded by the Hong Kong Research Grants Council.
REFERENCES
1. Davis, P. and Sommerville, F., Low-Cost Axle Load Determination,
Proceedings, 13th ARRB & 5th REAAA Combined Conference, 1986,
Part 6, p 142-149.
2. Peters, R.J., AXWAY - a System to Obtain Vehicle Axle Weights,
Proceedings, 12th ARRB Conference, 1984, 12 (2), p 10-18.
3. Peters, R.J., CULWAY - an Unmanned and Undetectable Highway Speed
Vehicle Weighing System, Proceedings, 13th ARRB & 5th REAAA
Combined Conference, 1986, Part 6, p 70-83.
4. Cebon, D. Assessment of the Dynamic Wheel Forces Generated by
Heavy Vehicle Road Vehicles. Symposium on Heavy Vehicle Suspension
Characteristics, ARRB, 1987.
5. Cantineni, R., Dynamic Behaviour of Highway Bridges Under The
Passage of Heavy Vehicles. Swiss Federal Laboratories for Materials
Testing and Research (EMPA) Report No. 220, 1992, 240p.
6. O'Connor, C. and Chan, T.H.T., Dynamic Wheel Loads from Bridge
Strains. Structural Engineering ASCE, 1 14 (STS), 1988, p. 1703- 1723.
7. Briggs, J.C. and Tse, M.K. Impact Force Identification using Extracted
Modal Parameters and Pattern Matching, International Journal of Impact
Engineering, 1992, Vol. 12, p361-372.
8. Bendat, J.S. and Piersol, J.S., Engineering Application of Correlation and
Spectral Analysis. John Wiley & Sons, Inc. Second Edition, 1993.
1097
1098
ESTIMATING THE BEHAVIOUR OF A
NONLINEAR EXPERIMENTAL MULTI DEGREE
OF FREEDOM SYSTEM USING A FORCE
APPROPRIATION APPROACH
P.A. Atkins J.R. Wright
Dynamics and Control Research Group
School of Engineering, Simon Building, University of Manchester,
Oxford Road, Manchester. M13 9PL
ABSTRACT
The identification of nonlinear multi degree of freedom systems involves a
significant number of nonlinear cross coupling terms, whether the identifi¬
cation is carried out in spatial or modal domains. One possible approach
to reducing the order of each identification required is to use a suitable
pattern of forces to drive any mode of interest. For a linear system, the
force pattern required to drive a single mode is derived using a Force Ap¬
propriation method. This paper presents a method for determining the
force pattern necessary to drive a mode of interest of a nonlinear system
into the nonlinear region whilst the response is controlled to remain in pro¬
portion to the linear mode shape. Such an approach then allows the direct
nonlinear modal terms for that mode to be identified using the Restoring
Force method. The method for determining the relevant force patterns is
discussed. The implementation of the method for experimental systems is
considered and experimental results from a two degree of freedom ’bench¬
mark structure’ are presented.
INTRODUCTION
Force Appropriation [1] is used in the analysis of linear systems to de¬
termine the force patterns which will induce single mode behaviour when
applied at the relevant natural frequency. This technique is used in the
aerospace industry during Ground Vibration Tests: each normal mode of
1099
a structure is excited using the derived force pattern and thus identified
in isolation. Current practice, when the presence of nonlinearity is sus¬
pected, is to increase input force levels and monitor the variation of tuned
frequencies. Some information about the type of nonlinearity present may
be found, but no analytical model can be derived. Thus predictions for
behaviour at higher levels of excitation axe not possible.
A number of techniques for identifying nonlinearity, for example the Restor¬
ing Force method [2], have been demonstrated on systems with low num¬
bers of degrees of freedom. Unfortunately, in practice, structures have a
large number of degrees of freedom, often with a high modal density. A
classical Restoring Force approach to the identification of such systems
could involve a prohibitive number of cross coupling terms. The ability to
treat each mode separately, by eliminating the effects of the cross coupling
terms, would thus.be advantageous. Subsequent tests could then evaluate
the cross coupling terms.
For these reasons it would be useful to extend Force Appropriation to the
identification of nonlinear systems. An approach has been developed [6]
that allows an input force pattern to be derived that will result in a non¬
linear response in the linear mode shape of interest. This force pattern is
derived using an optimisation approach. The mode of interest can then be
identified using a single degree of freedom nonlinear identification method.
In this work the Restoring Force method is used to examine the nonlinear
response of a particular linear mode and an application of this approach
to a two degree of freedom experimental system is presented.
THEORY
The theoretical approach is demonstrated for the two degree of freedom
system with spring grounded nonlinearity shown in figure 1. The equations
of motion for this system in physical space are:
m 0
0 m
+
+
(1 + a)c
—ac
'±1 1
—ac
(1 + d)c
\
X2 J
(l -}- Q^k
—ak
fill
—ak
(1 + a)k
l^x\
0
where is the cubic stifEhess coefficient and a is a constant that allows the
frequency spacing of the natural frequencies to be varied. These equations
can be transformed to linear modal space using the transformation:
{i} = [ij>]{u} (2)
where [(j>] is the modal matrix of the underlying linear system and the
vector {u} defines the modal displacements. For this symmetrical system
1100
the modal matrix is
1 1
1 -1
(3)
The equations of motion transformed to linear modal space using the nor¬
malised modal matrix are:
771 0 \ ill
0 771 (112
+
c 0
0 (1 + 2g)c
1^2/
■jb 0
Ull
0 (1 + 2a)k
1^2]
P{Ul-U2y/4:'
-U2)V4
where {p} is the modal input vector. It can be seen from the above equa¬
tions that the cubic nonlinearity couples the modes in linear mod^ space;
in fact there are a significant number of terms for a single nonlinearity.
The proposed method aims to determine the force pattern that will reduce
the response of this system to that of a single mode.
It was shown in a previous paper [3] that this can be achieved by seeking a
force vector that will cause motion only in the target mode, by eliminating
motion in the coupled mode. In practice, physical data from transducers
are available. Any subsequent transformations would be time consuming.
It is shown below that causing motion in one mode to be zero is equiva¬
lent to forcing motion in a linear mode shape, mode one in this example.
Consider the coordinate transformation {a;} = [<^]{ii} or more explicitly
for the two degree of freedom system in Figure 1:
Thus
1
2
‘l -1
1 1
(5)
(6)
and enforcing the first mode shape (1, 1) in physical space should give a
second modal displacement of zero.
1
2
1 -1
1 1
1
1
(7)
So if an excitation is applied which causes the nonlinear system to vibrate
in its first linear mode shape, the response will be composed only of ui and
the influence of the coupled mode, U2, will then have been eliminated.
The method must therefore derive a force pattern which will cause the
system to vibrate in one of its linear mode shapes. It has been shown
1101
in a previous paper [3] that if the response contains harmonics then the
force pattern must also contain harmonics in order to control the harmonic
content of the response. In theory, the responses will be an infinite series
of harmonics, but this series is truncated in this case of a cubic stiffness
nonlinearity to include only the fundamental and third harmonic terms.
The physical input forces will thus be of the form:
fi(t) = Fii cos(a;ea;t 4- ^ii) 4- F13 cos(a;ea;t + (^13) (8)
f2{t) = F21 C0s(a;ea;t + (j>2l) 4* F23 COS{uJext + fe) (9)
where is the excitation frequency. Parameters for these force patterns
may then be chosen such that only mode one is excited.
OPTIMISATION APPROACH
In general, no a priori model of the system exists so an optimisation routine
is used to determine the force pattern parameters required to maximise the
contribution of the mode of interest. The objective function, the quantity
that the optimisation routine seeks to minimise, must be representative
of the deviation of the response from the target linear mode shape. The
objective function, F, that was chosen in this case was based on the vector
norm [4] of the two physical responses, Xi and X2, and is shown below:
where and 02 are elements of the mode shape vector for the target
mode. This summation is carried out over one cycle of the fundamental
response. The number of data points per cycle is npts and Xki the kth
response at the itk sample. This objective function allows the response
to contain harmonics and can be extended to more degrees of freedom by
choosing a reference displacement and subtracting further displacements
from it. The Variable Metric optimisation method [5] was used in this
work as it has been found to produce the best results for simulated data.
The application of this method to a two degree of freedom system such
as that shown in figure 1 is detailed in [6]. Optimised force patterns are
obtained at several levels of input amplitude. These force patterns are then
applied and the Restoring Force method is used to curve fit the resulting
modal displacement and velocity time histories to give the direct linear and
nonlinear coefficients for the mode of interest.
1102
EXPERIMENTAL IMPLEMENTATION
The simulated application of this method assumed that certain parame¬
ters were known. In order to carry out an identification of an experimental
structure, these parameters must be measured or calculated. Some pro¬
cessing of experimental data is necessary in order to apply the Restoring
Force method. The restoring force of a system can be expressed for a single
degree of freedom system as:
h{x, x) = f(t) - mx (11)
where h{x, x) is the restoring force and f{t) the input force. A similar ex¬
pression applies to the modal restoring force for an isolated mode. Thus the
input force, acceleration, velocity and displacement must be calculated at
the each time instant. A similar expression applies to the modal restoring
force for an isolated mode. In the experimentaJ case it is usual to measure
acceleration and input force; the remaining two states must therefore be
obtained by integration of the acceleration time history. Frequency domain
integration [7] was used for this purpose. High pass filtering was used to
remove any low frequency noise which can be amplified by this type of
integration. Several methods have been suggested for estimating modal
mass, but in this study a method developed by Worden and Tomlinson [8]
was used. An estimate for the modal mass is obtained and then an error
term is included in the curve fit which will iteratively yield a more accu¬
rate estimate of the mass. Generally the mass value will converge after one
iteration.
The objective function used in the simulations was calculated from the
displacement time histories. In the experimental case, acceleration was
used rather than displacement as it was considered that using ’raw’ data
would be quicker and give less opportunity for error. In the simulated
case, the system parameters were known a priori so the modal matrix
of the underlying linear system could be calculated. For most types of
nonlinearity the response of the system at low input force levels will be
dominated by linear terms. Normal mode tuning [1] was therefore applied
at low force levels to yield an approximation to the modal matrix of the
underlying linear system.
A quality indicator to give some idea of the effectiveness of the optimisa¬
tion performed would be advantageous. Results corrupted by background
noise, for example, could then be discarded. A perfect optimisation will
occur when the ratio of measured accelerations exactly matches the mode
shape ratio specified for the mode of interest. Thus a least squares fit of
the sampled accelerations was carried out over a cycle of the fundamental
frequency and the percentage error of the measured mode shape to the
1103
required mode shape was calculated. This percentage error will indicate
whether the optimisation has been successful.
To assess the accuracy of the parameters estimated using this method,
an identification was carried out using a conventional Restoring Force ap¬
proach in physical space. A band limited random excitation was used, and
the physical data processed and curve fitted. The physical parameters were
then transformed to modal space. The direct linear and nonlinear param¬
eters for modes one and two are shown in table 1. It should be noted that
although this conventional Restoring Force approach is possible for this
two degree of freedom system, it will not generally be possible since the
number of terms in the curve fit increases dramatically when different t3q)es
of nonlinearity and more degrees of freedom are included. It is carried out
in this case as a means of validating the proposed method.
EXPERIMENTAL SETUP
The rig constructed consisted of two masses on thin legs connected in series
by a linear spring, each mass being driven by a shaker. A cubic nonlin¬
earity was introduced between the first mass and ground using a clamped-
clamped beam attached at the centre which will yield a cubic stiffness for
large deflections [9], A schematic diagram of the rig is shown in figure 2.
The force input by each shaker was measured using a force gauge and the
acceleration of each mass was measured using an accelerometer in the po¬
sitions also shown in figure 2, Acceleration and force data were acquired
using a multiple channel acquisition system, the optimisation routine was
carried out on line.
RESULTS
Normal mode tuning of the rig gave natural frequencies of 20.67 Hz and
24.27 Hz and a modal matrix of:
3.87 5.03
5.52 -3.27
(12)
The excitation frequency was chosen to be slightly lower than the natural
frequency of the mode of interest in order to avoid the problems associated
with force drop out which are worst at the natural frequency. For each
mode optimisation was performed at three input force levels, the highest
level was as high as possible so as to excite the nonlinearity strongly. The
optimisation routine was carried out using the voltage input into the signal
generator as the variable. The force input into the structure was measured
1104
for use in the Restoring Force identification but was not used in the opti¬
misation as it is not directly controllable. The details of the optimisation
for each force level are presented in table 2. The optimised forces and re¬
sulting accelerations for mode one are shown in figure 3 and figure 4. The
acceleration data for the optimised force patterns were then integrated and
the modal restoring force for the mode of interest calculated. The initial
estimate of modal mass for the calculation of the modal restoring force was
tahen from a previous paper [10] in which the rig was identified using a
using a physical parameter identification method. The mass was estimated
in this paper to be 2.62 kg, this physical mass will then be equal to the
modal mass since the modal matrix was normalised to be orthonormal.
The restoring force data was then transformed to modal space. The modal
restoring force surface obtained using optimised force inputs for mode one
is shown in figure 5 and a stiffness section through this surface is shown in
figure 6. The restoring force time histories were then curve fitted against
modal velocity and displacement.
The estimated parameters for mode one axe shown in table 3. It can be
seen that they do not compare very well with those estimates obtained
using the band limited random excitation. It was suspected that at lower
excitation levels the estimates were being distorted by linear dependence
[11]. Linear dependence is a problem which occurs when curve fitting a
harmonic response from a linear system; the equations of motion may be
identically satisfied by mass and stiffness terms modified by an arbitrary
constant. This condition is avoided by the harmonic terms introduced into
the response by nonlinearity. The curve fit was thus repeated using only the
data obtained from the highest level of excitation; the estimates obtained
are shown in table 4. It can be seen that the linear parameter estimates
now agree well with the band limited random results. The estimates for
the cubic stiffness coefficient do not appear to agree so well. The standard
deviation on the cubic stiffness derived from the band limited random
excitation is approximately a third of the value of the parameter itself. The
uncertainty on this parameter occurs because the nonlinearity is not very
strongly excited by this type of excitation. A stiffness section through the
restoring force surface, figure 7, shows little evidence of a cubic stiffness
component. If a higher level of excitation were possible then a better
estimate may be achieved.
The identification w^ repeated for mode two. The restoring force surface
obtained and a stiffness section through it are shown in figure 8 and figure 9.
It can be seen from the stiffness section that the nonlinearity is not very
strongly excited. The estimated direct modal parameters are shown in
table 5. It can be seen that these results agree quite well with those
obtained using band limited random excitation. It is considered that the
discrepancy between the two sets of results, in particular the mass and
1105
stiifiiess estimates, is again due to linear dependence.
CONCLUSIONS
An extension of the force appropriation method has been proposed for
nonlinear systems. In this method, an optimisation routine is used to
determine the force patterns which will excite a single mode nonlinear
response . The direct linear and nonlinear modal parameters can then be
estimated from a curve fit of the modal restoring force surface. The method
was applied to an experimental two degree of freedom system whose modes
were coupled in linear modal space by a spring grounded nonlinearity.
A conventional restoring force identification was performed using a band
limited random signal for comparison. The parameters estimated from the
single mode responses were found to agree quite well with those from the
band limited random tests.
REFERENCES
1. Holmes P., Advanced Applications of Normal Mode Testing, PhD
Thesis, University of Manchester 1996.
2. Hadid M.A. and Wright J.R., Application of Force State Mapping
to the Identification of Nonlinear Systems Mechanical Systems and
Signal Processing, 1990, 4(6), 463-482
3. Atkins P.A., Wright J.R., Worden K., Manson G.M. and Tomlinson
G.R., Dimensional Reduction for Multi Degree of Freedom Nonlinear
Systems, International Conference on Identification in Engineering
Systems 1996, 712-721
4. Kreyszig E., Advanced Engineering Mathematics, Wiley, 6th Edition
5. Press W.H., Teukolsky S.A., Vettering W.T. and Flannery B.P., Nu¬
merical Recipes in Fortran Cambridge University Press, 6th Edition
6. Atkins P.A. and Wright J.R., An Extension of Force Appropriation
for Nonlinear Systems Noise and Vibration Engineering, Proceedings
of ISMA21(2), 915-926, 1996
7. Worden K., Data Processing and Experiment Design for the Restor¬
ing Force Method, Part I: Integration and Differentiation of Measured
Time Data, Mechanical Systems and Signal Processing, 4(4) 295-319,
1990
1106
8. Ajjan Al-Hadid M., Identification of Nonlinear Dynamic Systems us¬
ing the Force State Mapping Technique, PhD Thesis, Queen Mary
College, University of London, 1989
9. Storer D.M., Dynamic Analysis of Nonlinear Structures Using Higher
Order Frequency Response Function, PhD Thesis, University of Manch¬
ester, 1991
10. Atkins P. and Worden K., Identification of a Multi Degree of Freedom
Nonlinear System, Proc. of IMAC XV, 1997
ACKNOWLEDGEMENTS
This work was supported by E.P.S.R.C. under research grant number
GR/J48238 at the University of Manchester
Modal parameter
Mode one
Mode two
k (N/m)
c (Nm/s)
/3(N/m^)
m (kg)
4.87 X 10^
10.11
3.83 X 10®
2.60
6.49 X 10^
9.49
8.90 X 10*
3.06
Table 1: Direct modal parameters estimated from curve fit of band limited
random data
Low forcing
Medium forcing
High forcing
Fii (Volts)
1.0
2.0
3.0
F21 (Volts)
1.0
2.0
3.0
u)^ (Hz)
20.0
20.0
20.0
initial mode shape ratio
0.72
0.68
0.65
final mode shape ratio
0.70
0.70
0.70
target mode shape ratio
0.70
0.70
0.70
percentage error
0.14
0.03
0.71
Table ,2: Details of optimisation for mode one
Table 3: Direct modal parameters estimated from optimised responses
Model parameter Estimated parameter
k (N/m) 4.57 X 10^
c (Nm/s) 8.62
/?(N/m3) 6.81 X 10®
m (kg) _ _ 2^75 _
Table 4: Direct modal parameters estimated using high force level only
Model parameter
Estimated parameter
k (N/m)
c (Nm/s)
m (kg)
5.37 X 10^
10.16
2.45 X 10®
2.29
Table 5: Direct modal parameters estimated for mode two
Figure 1: Two degree of freedom system
1108
0 s 0.40008
Figure 3: Optimised forces for mode one at a high force level
Figure 4: Accelerations responses to optimised forces
Figure 5: Modal restoring force surface for mode one
Force (N)
Figure 7: Stiffness section through modal restoring force derived from ran¬
dom excitation for mode one
1111
Figure 8: Modal restoring force surface for mode two
Figure 9: Stiffness section through modal restoring force for mode two
1112
POWER FLOW TECHNIQUES II
THE OPTIMAL DESIGN OF NEAR-PERIODIC STRUCTURES TO
MINIMISE NOISE AND VIBRATION TRANSMISSION
R.S. Langley, N.S. Bardell, and P.M. Loasby
Department of Aeronautics and Astronautics
University of Southampton
Southampton SO 17 IBJ, UK
1. INTRODUCTION
An engineering structure is said to be of "periodic" construction if a basic
structural unit is repeated in a regular pattern . A beam which rests on
regularly spaced supports is one example of a one-dimensional periodic
structure, while an orthogonally stiffened cylinder is an example of a two-
dimensional periodic structure. It has long been known that perfectly periodic
structures have very distinctive vibration properties, in the sense that "pass
bands" and "stop bands" arise: these are frequency bands over which elastic
wave motion respectively can and cannot propagate through the structure [1,2].
If the excitation frequency lies within a stop band then the structural response
tends to be localised to the immediate vicinity of the excitation source.
Conversely, if the excitation frequency lies within a pass band then strong
vibration transmission can occur, and it is generally the case that the resonant
frequencies of the structure lie within the pass bands.
Much recent work has been performed concerning the effect of random
disorder on a nominally periodic structure (see for example [3-5]). It has been
found that disorder can lead to localisation of the response even for excitation
which lies within a pass band, and this reduces the propensity of the structure
to transmit vibration. This raises the possibility of designing disorder into a
structure in order to reduce vibration transmission, and this possibility was
briefly investigated in reference [6] for a one-dimensional periodic waveguide
which was embedded in an otherwise infinite homogeneous system. The
present work extends the work reported in reference [6] to the case of a finite
near-periodic beam system, which more closely resembles the type of
optimisation problem which is. likely to occur in engineering practice. The
beam is taken to have N bays, and the design parameters are taken to be the
individual bay lengths. Both single frequency and band-limited excitation are
considered, and two objective functions are investigated: (i) the response in a
bay which is distant from the applied loading (minimisation of vibration
transmission), and (ii) the maximum response in the structure (minimisation
of maximum stress levels). In each case the optimal configuration is found by
employing a quasi-Newton algorithm, and the physical features of the resulting
design are discussed in order to suggest general design guidelines.
1113
2. ANALYTICAL MODEL OF THE NEAR-PERIODIC BEAM
2.1 Calculation of the Forced Response
A schematic of an N bay near-periodic beam structure is shown in Figure 1.
The structure is subjected to dynamic loading, and the aim of the present work
is to find the optimal design which will minimise a prescribed measure of the
vibration response. No matter what type of optimisation algorithm is
employed, this type of study requires repeated computation of the system
dynamic response as the design parameters are varied, and it is therefore
important to employ an efficient analysis procedure. In the present work the
h-p version of the finite element method (FEM) is employed: with this
approach the structure is modelled as an assembly of elements which have
both nodal and internal degrees of freedom. Each element has two nodes and
the nodal degrees of freedom consist of the beam displacement and slope; the
internal degrees of freedom are generalised coordinates which are associated
with a hierarchy of shape functions which contribute only to the internal
displacement field of the element. The internal shape functions used here are
the K-orthogonal Legendre polynomials of order four onwards - full details of
the present modelling approach are given in reference [7].
For harmonic excitation of frequency co the equations of motion of the
complete beam structure can be written in the form
where M and K are the global mass and stiffness matrices (assembled from the
individual element matrices taking into account the presence of any mass or
spring elements and allowing for constraints), q contains the system
generalized coordinates, F is the generalized force vector, and r| is the loss
factor, which in the present study is taken to be uniform throughout the
structure.
Equation (1) can readily be solved to yield the system response q. In the
present work it is convenient to use the time averaged kinetic and strain
energies of each of the N bays as a measure of the response - for the nth bay
these quantities can be written as and say, where
r,=(£0V4)9;>„?„. (2.3)
Here and K„ are the mass and stiffness matrices of the nth bay, and q„ is
the vector of generalized coordinates for this bay.
Many of the physical features of the forced response of a near-periodic
structure can be explained in terms of the free vibration behaviour of the
1114
associated perfectly periodic structure. The following section outlines how the
present finite element modelling approach can be used to study the pass bands
and stop bands exhibited by a perfect periodic structure.
2.2 Periodic Structure Analysis
The finite element method described in section 2.1 can be applied to a single
bay of a perfectly periodic structure to yield an equation of motion in the form
Dq=F, D=-coW--(l+/r|)ii:, (^’5)
where the matrix D is referred to as the dynamic stiffness matrix. In order to
study wave motion through the periodic system it is convenient to partition D,
q and F as follows
r
Du
ft:
D=
D„
D,r
. F=
0
Dr,
F„
V V
(6-8)
where L relates to the coordinates at the left most node, R relates to those at
the right most node, and / relates to the remaining "internal" coordinates.
Equations (4-8) can be used to derive the following transfer matrix relation
between the displacements and forces at the left and right hand nodes
V /
V /
(9,10)
Equation (9) can now be used to analyze wave motion through the periodic
system: such motion is governed by Bloch’s Theorem, which states that
i^L F i)=exp{-ie-b){qii -F^) where 8 and 5 are known respectively as the phase
and attenuation constants. A pass band is defined as a frequency band over
which 6=0, so that wave motion can propagate down the structure without
attenuation. It follows from equation (9) that
(T-Ie
-ie-6'
( \
0
F,
io
V
V y
(11)
SO that 8 and 6 can be computed from the eigenvalues of T, thus enabling the
pass bands and stop bands to be identified.
1115
2.3 Optimisation Procedure
Equations (l)-(3) enable the forced response of the system to be calculated for
any prescribed set of system properties. The aim of the present analysis is to
compute the optimal set of system properties for a prescribed design objective,
and in order to achieve this equations (l)-(3) are evaluated repeatedly as part
of an optimisation algorithm. As an example, it might be required to minimise
the kinetic energy of bay N by changing the various bay lengths. In this case
equations (l)-(3) provide the route via which the objective function (the kinetic
energy in bay N) is related to the design parameters (the bay lengths), and the
optimisation algorithm must adjust the design parameters so as to minimise the
objective function. The optimisation process has been performed here by using
the NAg library routine E04JAF [8], which employs a quasi-Newton algorithm.
This type of algorithm locates a minimum in the objective function, although
there is no indication whether this minimum is the global minimum or a less
optimal local minimum. The probability of locating the global minimum can
be increased significantly by repeated application of the NAg routine using
random starts, i.e. random initial values of the design parameters. Numerical
investigations have led to the use of 30 random starts in the present work.
3. NUMERICAL RESULTS
3.1 The System Considered
The foregoing analysis has been applied to a beam of flexural rigidity El, mass
per unit length m, and loss factor r|=0.015, which rests on A+1 simple
supports, thus giving an A’-bay near-periodic system. The design parameters
are taken to be the bay lengths (i.e. the separation of the simple supports), and
the design is constrained so that the length of any bay lies within the range
0.9L^,<1.1L, where L, is a reference length. A non-dimensional frequency
Q is introduced such that 0.=(oL,N{m/Er), and the non-dimensional kinetic and
strain energies of a bay are defined as T,'-T,^(EI/L^^\F\^) and
U,'~U„iEI/L,^\F\'^) where F is the applied point load. As discussed in the
following subsections, two objective functions are considered corresponding
to minimum vibration transmission and minimum overall response. In all
cases the excitation consists of a point load applied to the first bay and the
response is averaged over 1 1 equally spaced point load locations within the
bay. For reference, the propagation constants for a periodic system in which
all the bay lengths are equal to L, are shown in Figure 2 - the present study is
focused on excitation frequencies which lie in the range 23<f2<61, which
covers the second stop band and the second pass band of the periodic system.
3.2 Design for Minimum Vibration Transmission
In this case the objective function is taken to be the kinetic energy in bay N,
so that the aim is to minimise the vibration transmitted along the structure.
Three types of loading are considered; (i) single frequency loading with Q=50,
1116
which lies within the second pass band of the ordered structure; (ii) band-
limited loading with 40<Q<60, which covers the whole of the second pass
band; (iii) band-limited loading with 23<Q<61, which covers the whole of the
second stop band and the second pass band.
Results for the optimal design under single frequency loading are shown in
Table 1; in all cases it was found that the bay lengths were placed against
either the upper bound (U=1.1L,) or the lower bound (L=0.9L,), and significant
reductions in the energy level of bay N were achieved. In this regard it should
be noted that the dB reduction quoted on Table 1 is defined as -101og(r;v/^;^r)
where is the kinetic energy in the final bay of the ordered system. The
optimal designs shown in Table 1 all tend to consist of a bi-periodic structure
in which the basic unit consists of two bays in the configuration LU. The pass
bands and stop bands for this configuration are shown in Figure 3, and further,
for the optimal 12 bay system is shown in Figure 4 over the frequency
range 0<Q<250. By comparing Figures 3 and 4 it is clear why the selected
design is optimal - the new bi-periodic system has a stop band centred on the
specified excitation frequency Q=50. It can be seen from Figure 4 that the
improvement in the response at the specified frequency 0=50 is accompanied
by a worsening of the response at some other frequencies.
Results for the optimal design under band-limited excitation over the range
40<Q<60 are shown in Table 2. In some cases two results are shown for the
optimised "Final Energy": in such cases the first result has been obtained by
forcing each bay length onto either the upper (U) or lower (L) bound, while
the second result has been obtained by using the NAg optimisation routine.
If only one result is shown then the two methods yield the same optimal
design. The "bound" result is easily obtained by computing the response under
each possible combination of U and L bay lengths - this requires 2^ response
calculations, which normally takes much less CPU time than the NAg
optimisation routine. It is clear from Table 2 that the additional improvement
in the response yielded by the full optimisation routine is minimal for this
case. The response curve for the 12-bay system is shown in Figure 5, where
it is clear that a significantly reduced response is achieved over the specified
frequency range; as would be expected an increase in the response can occur
at other frequencies. It is interesting to note that most of the optimal designs
shown in Table 2 lack symmetry - however, it follows from the principle of
reciprocity that a design which minimises vibration transmission from left to
right will also minimise transmission from right to left. It should therefore be
possible to "reverse" the designs without changing the transmitted vibration
levels. This hypothesis is tested in Figure 6 for a 12 bay structure - the figure
shows the energy distribution for the optimal design UUULUULLLLLU and
for the reversed design ULLLLLUULUUU. Although the detailed distribution
of energy varies between the two designs, the energy levels achieved in bay
1117
12 are identical, as expected.
Results for the optimal design under wide-band excitation 23<Q<61 are shown
in Table 3, and the response curve for the 12-bay optimised system is shown
in Figure 7. The form of optimal design achieved is similar to that obtained
for the narrower excitation band 40<Q<60, although there are detailed
differences between the two sets of results. In each case there is a tendency
for a group of lower bound bays (L) to occur in the mid region of the
structure, and a group of upper bound bays (U) to occur at either end. This
creates an "impedance mismatch" between the two sets of bays, which
promotes wave reflection and thus reduces vibration transmission along the
structure. By comparing Tables 1-3, it is clear that the achievable reduction
in vibration transmission reduces as the bandwidth of the excitation is
increased.
3.3 Design for Minimum ''Maximum” Strain Energy
In this case the strain energy U„ of each bay is computed and the objective
function is taken to be the maximum value of U„. As a design objective, this
procedure can be likened to minimising the maximum stress in the structure.
As in the previous section the three frequency ranges Q=50, 40<n<60, and
23<Q<61 are considered, and the present study is limited to systems having
9,10, 11, or 12 bays; the optimal designs achieved are shown in Table 4.
Considering the single frequency results (Q.=50) shown in Table 4, it is clear
that a large dB reduction is achieved only for those systems which have an
even number of bays; furthermore, the optimal energy obtained has the same
value (0.0297) in all cases. This can be explained by noting that for an odd
number of bays the frequency Q=50 lies near to an anti-resonance of the
ordered structure, whereas a resonance is excited for an even number of bays -
this feature is illustrated in Figure 8 for the 12 bay structure. The repeated
occurrence of the optimal energy 0.0297 arises from the fact that the initial bay
pattern ULLLUUU occurs in all four designs - it has been found that this
pattern causes a vibration reduction of over 20dB from bay 1 to bay 8, so that
the response in bay 1 (the maximum response) is insensitive to the nature of
structure from bay 8 onwards.
The optimal "bounded" designs arising for band-limited excitation either tend
to be of the "UL" bi-periodic type or else nearly all the bays are assigned the
same length. However it should be noted that in all cases the design produced
by the NAg optimisation routine offers an improvement over the "bounded"
design, particularly for the wide-band case (23<n<61). It is clear from Table
4 that the achieved reduction in strain energy reduces as the bandwidth of the
excitation is increased.
1118
4. CONCLUSIONS
The present work has considered the optimal design of a near-periodic beam
system to minimise vibration transmission and also maximum stress levels.
With regard to vibration transmission it has been found that very significant
reductions in transmission are achievable with relatively minor design changes.
The optimum design normally involves placing the design parameters (the bay
lengths) on the permissible bounds, and this means that a simple design search
routine can be used in preference to a full optimisation algorithm. With regard
to minimum stress levels, it has been found that the optimal design for wide¬
band excitation is not normally a "bounded" design, and thus use of a full
optimisation algorithm is preferable for this case. For both vibration
transmission and maximum stress levels, the benefits obtained from an optimal
design decrease with increasing excitation bandwidth, but nonetheless very
significant reductions can be obtained for wide-band excitation.
REFERENCES
1. S.S. MESTER and H. BENAROYA 1995 Shock and Vibration 2, 69-
95. Periodic and near-periodic structures.
2. D.J. MEAD 1996 Journal of Sound and Vibration 190, 495-524. Wave
propagation in continuous periodic structures: research contributions
from Southampton 1964-1995.
3. C.H. HODGES 1982 Journal of Sound and Vibration 82, 411-424.
Confinement of vibration by structural irregularity.
4. D. BOUZIT and C. PIERRE 1992 Journal of Vibration and Acoustics
114, 521-530. Vibration confinement phenomena in disordered, mono-
coupled, multi-span beams.
5. R.S. LANGLEY 1996 Journal of Sound and Vibration 189, 421-441.
The statistics of wave transmission through disordered periodic
waveguides.
6. R.S. LANGLEY 1995 Journal of Sound and Vibration 188, 717-743.
Wave transmission through one-dimensional near periodic structures:
optimum and random disorder.
7. N.S. BARDELL, R.S. LANGLEY, J.M. DUNSDON and T. KLEIN
1996 Journal of Sound and Vibration 197, 427-446. The effect of
period asymmetry on wave propagation in periodic beams.
8. ANON 1986 The NAg Fortran Workshop Library Handbook - Release
1. Oxford: NAg Ltd.
1119
TABLE 1
Optimal design of 1-D beam structure, to minimise energy transmission, D.=50.
Original Energy; Non-dimensional kinetic energy in bay N of the periodic structure.
Final Energy; Non-dimensional kinetic energy in bay N of the optimised structure.
No. of
Optimal Pattern
Original
Final
Reduction
Bays, N
Energy
Energy
(dB)
4
UULU
0.276E 1
0.804E-3
35.348
ULULU
0.609E-1
0.179E-3
■S
UULULULU
0.674E 0
0.613E-5
ULULULULU
0.564E-1
0.135E-5
46.216
1—
10
UULULULULU
0.424E 0
0.532E-6 ^
11
ulululululu
0.535E-1
0.n7E-6
56.604
12
UULULULULULU
0.289E 0
0.461E-7
67.966
13
ULULULULULULU
0.502E-1
O.lOlE-7
66.950
16
UULULULULULULULU
0.154E0
0.346E-9
_ _ —
86.484
17
ululululululululu
0.43 lE-1
0.761E-10
87.529 1
1120
TABLE 2
Optimal design of 1-D beam structure, to minimise energy transmission, 40<Q<60.
Original Energy: Non-dimensional kinetic energy in bay N of the periodic structure.
Final Energy: Non-dimensional kinetic energy in bay N of the optimised structure.
No. of
Optimal pattern
Original |
Final
Reduction
Bays, N
Energy
Energy
(dB)
■■
ULLU
0.670E 0
0.103E-1
18.112
5
ULLLU
0.63 IE 0
0.735E-2
19.338
0.711E-2
19.482
6
UULLLU
0.221E-2
22.407
7
ULLUULU
0.463E 0
0.171E-2
24.335
8
UULLLLLU
0.430E 0
0.966E-3
26.487
0.914E-3
26.725
9
UUULLLLLU
0.444E 0
0.341E-3
31.142
10
UUUULLLLLU
0.449E 0
0.192E-3
33.681
0.189E-3
33.758
11
ULLUUUULLLU
0.291E0
0.821E-4
35.504
12
UUULUULLLLLU
0.201E0
0.352E-4
37.558
13
1 ULUUUUULLLLLU
0.199E0
0.153E-4
41.148
1121
TABLE 3
Optimal design of 1-D beam structure, to minimise energy transmission, 23<Q<6L
Original Energy; Non-dimensional kinetic energy in bay N of the periodic structure.
Final Energy: Non-dimensional kinetic energy in bay N of the optimised structure.
6
LLLLUU
7
LLLULUU
8
LLLLLUUU
9
ULLLLLLUU
0.494E 0
0.183E0
0.175E0
0.648E-2
0.246E-2
0.180E-2
III »
UUULLLLLUU
0.105E0
0.277E-3
2.762
3.916
18.821
18.715
19.878
21.868
25.787
11
UUULLLLLLUU
0.105E0
0.776E-4
31.313
12
UlTULLLLLLLUU
0.166E0
0.526E-4
34.991
13
UUULLLLLLULUU
0.973E-1
0.282E-4
35.379
14
UUUULLLLLLULUU
0.581E-1
0.122E-4
36.778
TABLE 4
Optimal design ofl-D beam structure, to minimise "maximum” strain energy.
Bay No.: Bay in which the optimal minimum “maximum” non-dimensional strain energy occurs
Original Energy: Initial “maximum” non-dimensional bay strain energy of the periodic structure.
Final Energy: Non-dimensional strain energy in bay N’ of the optimised structure.
No of
Optimal
Original
Final
Bay
Reduction
Bays, N
Pattern
Energy
Energy
No., N’
(dB)
Q.=50
9
ULLLUUUUL
0.667E-1
0.297E-1
1
3.514
0.296E-1
1
3.528
10
ULLLUUUULU
0.540E 0
12
ULLLULUULULU
0.404E 0
0.297E-1
1
11.336
0.296E-1
1
11.351
40<Q<60
9
UUUUULULU
0.486E 0
0.710E-1
1
8.354
0.449E-i
1-2
10.344
10
ULULULULLL
0.606E 0
0.643E-1
1
9.743
0.45 IE- 1
1-2
11.283
11
ULULULULLUU
0.456E 0
0.682E-1
I
8.252
0.425E-1
1-2
10.306
12
UUUIJUUUXJUUUL
0.332E 0
0.550E-1
2
7.808
0.412E-1
1-2
9.062
23<n<61
9
LLLLLLLLL
0.234E 0
0.203E 0
1
0.617
0.979E-1
1
3.784
n
LLLLLLLLLL
0.200E 0
0.178E0
1
0.506
■I
0.95 IE- 1
1-2
3.228
11
UUUUUUUUULL
0.198E0
0.193E0
1
O.lll
0.910E-1
1-2
3.376
12
UUUUUOUIJULUU
0.314E0
0.182E0
I
2.369
0.803E-1
1-2
5.922
1123
Phase £ AUenualion
Figure 1; A simply supported periodic
Itlijsj
kinelic.enorgy in bay 12, T,
1128
EFFECTS OF GEOMETRIC ASYMMETRY ON VIBRATIONAL
POWER TRANSMISSION IN FRAMEWORKS
J L Homer
Department of Aeronautical and Automotive Engineering
and Transport Studies, Loughborough University
Loughborough, Leics , LEll 3TU, UK
ABSTRACT
Many sources, such as machines, are installed on supports, or frameworks,
constmcted from beam-like members. It is desirable to be able to predict
which wave types will be present at particular points in the support structure.
By using the concept of vibrational power it is possible to compare the
contributions from each wave type. Wave motion techniques are used to
determine the expressions for vibrational power for each of the various wave
types present. The results from the analysis show the amount of vibrational
power carried by each wave type and the direction of propagation.
Consideration is given to the effect on the vibrational power transmission of
introducing misalignment of junctions in previously symmetric framework
structures. By splitting a four beam junction in to, say, a pair of three beam
junctions separated by a small distance, it is possible to establish the effects of
separating the junctions on the various transmission paths. Unlike other
techniques using vibrational power to analysis frameworks, the model keeps
the contributions from each of the various wave types separate. This allows
decisions to be made on the correct vibrational control techniques to be
applied to the structure.
INTRODUCTION
When attempting to control vibration levels transmitted from a machine
through the various connections to the structure upon which it is mounted, it is
desirable to be able to identify and quantify the vibration paths in the stmcture.
Often large machinery installations are installed on frameworks consisting of
beam like members. These frameworks are then isolated from the main
structure. Simple framework models are also used in the initial design stages
of automotive body shell structures to determine dynamic responses.
If the dominant transmission path in the framework is identified it is
possible to reduce vibration levels by absorbing the mechanical energy along
the propagation path in some convenient manner. By utilising the concept of
vibrational power it is possible to quantitatively compare the wave type
contributions to each transmission patL In order to predict vibrational power
transmission in a framework, it is necessary to identify the wave amplitude
reflection and transmission coefficients for each joint in the structure. Lee and
Kolsky [1] investigated the effects of longitudinal wave impingement on a
junction of arbitrary angle between two rods. Similarly Doyle and Kamle [2]
examined the wave amplitudes resulting from a flexural wave impinging on
the junction between two beams. By using the reflection and transmission
coefficients for different joints, it is possible to predict the vibrational power
associated with flexural and longitudinal waves in each section of the
1129
framework. Previous investigations [3,4] have considered the effects of bends
and junctions in infinite beams. This work was extended to consider the finite
members which constitute frameworks [5]. Unlike other techniques [6, 7]
utilising energy techniques to analyse frame-works, the technique produces
power distributions for each wave type present in the structure. By comparing
the results for each wave type, it is possible to apply the correct methods of
vibration control.
The technique is used to investigate the effect of geometric asymmetry
on the vibration transmission, due to steady state sinosodial excitation, in a
framework structure similar to, say, those used in the automotive industry
(figure 1). By splitting a four beam junction into a pair of three beam
junctions separated by a known distance, it is possible to establish the effect of
junction separation on the dominant transmission paths. The investigation
presented is limited to one dimensional bending waves and compressive waves
only propagating in the structure. To consider the addition of other wave
types ie. torsional waves and bending waves in the other plane, the analysis
presented here for the junctions should be extended as indicated by Gibbs and
Tattersall [3].
TRANSMITTED POWER IN A UNIFORM BEAM
For flexural wave motion, consider a section of a uniform beam carrying a
propagating flexural wave. Two loads act on this beam element, the shear
force and the bending moment. It is assumed that the flexural wave can be
described by using Euler-Bemoulli beam theory, so that the displacement can
be expressed as
W(x,t) = Af sin (cot-kfx),
the shear force acting on a section as
S = Eia3W/ax3,
and the bending moment on the section as
B = El 02 W/ax2.
Then the instantaneous rate of working X at the cross-section is given
by the sum of two terms (negative sign merely due to sign convention).
0t 0x0t 0X^ 0X^ 3x0t
The time averaged power
{P)f = (1/T) I X dt then is given by (P)f = Elkf coAf
I (1)
For longitudinal wave motion consider a section of a uniform beam
with a longitudinal wave propagating through the beam
U (x, t) = Ai sin (0)t-kix)
1130
The instantaneous rate of working X is then
X= -EA(au/ax)u
and the time averaged power is
Xdt = rEA0)kiA|
(2)
If dissipation is present in the structure, the modulus of elasticity may
be considered to be a complex quantity
= E(l + iTi)
where represents the loss factor of the material, present due to
inherent material damping.
The displacement of a beam at a distance x from the source, due to
flexural wave motion may now be considered to be, assuming that
material damping is small.
1 ^
-kri-
W = Afe 4
and the resulting time averaged power is given by
(P}f = EIcok^e-'‘'’i2 Af
(3)
The above reduces to equation (1) at the source.
Similarly, the displacement of beam, due to longitudinal wave motion
may be considered to be
T ^
-kiTi-
U = Aie ^ sin(cot-kix)
and the resulting time averaged longitudinal power may be rewritten as
(P),=iEAcok,e-‘''^’‘A?
(4)
WAVE TRANSMISSION THROUGH A MULTI BRANCH JUNCTION
Consider a four branch junction as shown in figure 2. Assuming only flexural
and longitudinal waves propagating in the structure, the displacements of Arm
1 will be, where A4 represents the impinging flexural wave arriving from
infinity.
Wi(x,t) = (Aie^f’"" + A3e‘^fi'' + A4e e‘“^
(5)
1131
(6)
U,(x,t) = (Aae''‘‘i’‘)e‘“‘
Similarly for arms 2 to 4 the displacement will be,
where ^ cos 0n and n is the beam number
W„(v„,t) = )e‘“‘
(V)
(8)
Here A3, A4, 64^ are travelling flexural wave amplitudes', Ai and
B2n are near field wave amplitudes and Aa and are travelling
longitudinal wave amplitudes.
In previous work [2] in this field a theoretical model was used in which
it was assumed that the junction between the beams was a rigid mass. The
mass or joint is modelled here as a section of a cylinder. This represents the
physical shape of most joints in practical systems. It has been shown [4] that
the joint mass has an insignificant effect on the reflected and transmitted
power for the range of values used in this work.
The joint mass Mj = pjTtL^J^/ 4, and the moment of inertia of the joint
isIj=ML2/8.
By considering the conditions for continuity and equilibrium at the
beam junction the following expressions may be written.
For each arm
For continuity of longitudinal displacement
L 3W
Ui=Un COsen-WnSinen + -^^sinen
For continuity of flexural displacement
W,=u„sinen+W„cos0„-| |^(l + cose„)
For continuity of slope
8W; ^ awn
1132
For the junction
Equilibrium of forces
' ' 2 ' ' 3x2 J
-vfp fax L32w„
1 1 I aVS 2 3v|;^
El Ai ^ + Mj ^ = i[E„ A„^cose„ +E„ I„^^sine„
3x J dt^ il^ 3V„ 3<
E T a^W, 3 r„, L3W|
El Ii — t^ + M; — T Wi- — — —
- ^ -I- iVi ; — y 1 “ “ ^T"
3x2 2 3x
n ;^TI ^ W
= Z En Ajj- ”Sin6n-EnIn 2*^ COS0n
1 I 5¥n
WAVE MOTION AT A FORCED OR FREE END
As indicated in figure 1 , the framework has one forced end and one free end.
Assuming the structure is only excited by a transverse harmonic force, the
boundary conditions are as follows:
at the forced end
E« = Pe^«t
3x2
EI^ = 0
3x2
EA^ = 0
Similarly at the free end the above boundary conditions apply with the
exception that
EI^ = 0
3x2
1133
POWER TRANSMISSION THROUGH A FRAMEWORK
The structure shown in figure 1 consits of one four-beam junction, two three-
beam junctions and four two-beam junctions. From the equations detailed in
the above two sections, it is possible to construct matrices of continuity and
equilibrium equations for sub structures. These may be combined to obtain
the overall matrix for the system. For the framework shown in figure 1, the
size of the overall matrix is 60x60. This matrix may be solved to obtain the
sixty unknown wave amplitude coefficients from which time averaged
transmitted power for each beam may be calculated using equations (3) and
(4).
Normalised nett vibrational power is then calculated at the centre of
each beam constituting the structure. Nett vibrational power may be
considered to be the difference between power flowing in the positive
direction and power flowing in the negative direction for each wave type.
Normalised nett power is considered to be nett power divided by total input
power. The input power to a structure may be calculated from the following
expression [8]
Input Power = ^ IFI IVI cos0
where 0 is the phase angle between the applied force and the velocity
of the structure at the forcing position.
Figures 3-6 show the nett normalised power in each arm of a
framework structure over the frequency range 0-lkHz excited by IN force,
whose material and geometric properties are given in Appendix 2. For the
results shown, angle 1 is 45® and angle x is 40® (or the ratio
= 0.89) and
L = y = 0.1m. Using these parameters the ratio of the length of beam No.6 to
beam No.4 is 0.12. The predicted flexural power is shown in figures 3 and 4
and from these it can be seen that the dominant transmission paths are arms 1
and 5, the forced and free arms. The transmitted power in arm 10 is next
dominant and comparable to arm 5 in the region 0-600Hz.
The response for ail other arms are small, typically less than 5% of
input power, with, as would be expected, arms 2 and 9 being approximately
identical in transmission properties.
Figures 5 and 6 show the nett normalised power for the longitudinal
waves in the structure. As the frequency range of interest corresponds to a
flexural Helmoltz number of 1 to 5 with L being the reference length, the
conversion of power from flexural to longitudinal waves is minimal. From the
figures it can be seen that beams 1, 5, 6 and 10 have identical transmission
characteristics, which would be expected at such large longitudinal
wavelengths. Significant longitudinal power is only observed in arms 3 and 8
in the frequency region 200-300Hz. This frequency region coincides with a
drop in the flexural power due to the structure being at resonance in that
region. It should be noted that power transmitted through arms 3 and 8 has
travelled through two junctions.
1134
EFFECT OF GEOMETRIC ASYMMETRY
By altering the ratio of angle 6i to angle 0x it is possible to alter the length of
beam 6 and hence move a pair of three arm junctions further or closer apart.
From the discussion in the previous section, it was seen, for the structure
under investigation, that the dominant flexural path, not surprisingly, is
through the centre of the structure, whilst the peaks in longitudinal power
occur in beams 3 and 8. Thus 0x was varied and the effect on transmission in
the dominant paths noted.
Figures 7-9 show flexural power for arms 5 and 10 and longitudinal
power for arm 8 for four values of 0x. The values chosen were 36°, 38.25°,
40° and 42.75° which are equivalent respectively to 0x over 0] ratios of 80%,
85%, 90% and 95%. Thus as 0x increases, the structure moves to being
symmetrical in nature. From figure 7, the increase in junction separation
decreases power in the frequency region 0-500Hz and increases it in the region
500-lkHz. In beam 10 (figure 10) the effect on the flexural power is reversed
with increase in junction separation leading to increased power below 500Hz
and decreased power above 500Hz. It should also be noted that increased
junction separation has little effect on the power below 250Hz. The effect was
also noted on all other beams which had both ends connected to a joint. It
may be concluded that at long flexural wavelengths the junction separation has
little effect with the impedance mis-match at the junctions being the important
criteria to effect transmission. It should also be noted that increasing power in
one arm ie. 5, causes a decrease in arms ie. 10, connected to it. An example of
the effect of junction separation on longitudinal power is shown in figure 9.
This shows nett normalised longitudinal power for arm 8 for the same
variation in 0x. Again minimal effect is seen at low frequencies, with
increased junction separation having different effects in different frequency
region. Increasing junction separation has little effect on the region between
200 and 300Hz when the longitudinal power was dominant. This would be
caused by the junction separation having little effect on the structures flexural
natural frequencies. Only by shifting those would the peaks in longitudinal
power by shifted in frequency.
CONCLUSIONS
Results are presented for normalised nett time average vibrational power for a
framework structure. The geometric symmetry of the structure is broken by
allowing one angle to decrease in value. The effects of varying the angle
change by up to 20% of its original value are investigated. Although the
results presented are for one example only, highlighted are the fact that
decreases in power in one part of the structure result in increases in power in
another part. Also shown was the effect of splitting a junction in to a pair of
junctions is minimal at low frequencies, or long wavelengths. From the results
of the analysis it is possible to establish frequencies and positions for
minimum power on the structure. Other configurations of framework
structure may be analysed by applying the equations presented.
1135
REFERENCES
1. J. P. LEE. and H. KOLSKY 1972 Journal of Applied Mechanics 39, 809-
813. The generation of stress pulses at the junction of two non-collinear
rods.
2. J. F. DOYLE and S. KAMLE 1987 Journal of Applied Mechanics 54,
136-140. An experimental study of the reflection and transmission of
flexural waves at an arbitrary T-Joint.
3. B. M. GIBBS and J. D. TATTERS ALL 1987 Journal of Vibration,
Acoustics, Stress and Reliability in Design, 109, 348-355. Vibrational
energy transmission and mode conversion at a corner junction of square
section rods.
4. J. L. HORNER and R. G. WHITE 1991 Journal of Sound and Vibration
147, 87-103. Prediction of vibrational power transmission through bends
and joints in beam-like structures.
5. J. L. HORNER 1994 Proceedings of the 5th International Conference on
Recent Advances in Structural Dynamics, SOUTHAMPTON UK, 450-
459. Analysis of vibrational power transmission in framework structures.
6. P. E. CHO and R. J. BERNHARD 1993 Proceedings of the 4th
International Congress on Intensity Techniques, SENLIS, France, 347-
354. A simple method for predicting energy flow distributions in frame
structures.
7. M. BESHARA and A. J. KEANE 1996 Proceedings of Inter-Noise '96,
LIVERPOOL, UK 2957-2962. Energy flows in beam networks with
complient joints.
8. R.J.PINNINGTON and R.G.WHITE 1981 Journal of Sound and Vibration
75, 179-197. Power flow through machine isolators to resonant and non-
resonant beams.
APPENDIX 1 - NOTATION
A
- Cross sectional area
Q
- Axial force
Af
- Amplitude of flexural wave
s
- Shear force
Ai
- Amplitude of longitudinal wave
T
- Time period
B
- Bending moment
t
- Time
E
- Young’s modulus
U
- Displacement due to
E*
- Complex Young’s modulus
longitudinal wave motion
F
- Excitation force
V
- Velocity
I
- Moment of inertia
w
- Displacement due to
flexural wave motion
Ij
Jw
- Moment of inertia of joint
X
- Instantaneous rate of
working
- Joint width
X
- Distance
- Loss factor
kf
- Flexural wave number
0n
- Angle of Arm n
ki
- Longitudinal wave number
Pj
- Joint density
L
- Joint length
M
- Moment force
Mj
n
P
- Joint mass
<}>
Phase angle
- Beam number
- Transverse force
¥n
- Distance along Arm n
1136
<P>f - Time averaged flexural power co - Frequency (rad/s)
<P>1 - Time averaged longitudinal power
APPENDIX 2 - MODEL PROPERTIES
33mm
6mm
5GN/m2
1180kg/m3
0.001
1
Beam Breadth
Beam Depth
Youngs Modulus
Density
Loss factor
Figure 1 : Framework Structure
iNett Normalised Power
Nett Normalised Power
Figure 5: Longitudinal Power - Beams 1-5
(Beam 1 - , Beam 2 . , Beam 3 . . Beam 4 - , Beam 5
Figure 6: Longitudinal Power - Beams 6-10
(Beam 6 - , Beam 7 . , Beam 8 . . Beam 9 - , Beam 10
1139
THE INFLUENCE OF THE DISSff ATION LAYER ON ENERGY
FLOW IN PLATE CONNECTIONS
Marek Iwaniec, Ryszard Panuszka
Technical University of Mining and Metallurgy,
Structural Acoustics and Intelligent Materials Group
30-059 Cracow, al. Mickiewicza 30, Poland
1. Introduction
Dynamic behaviour of mechanical strucmres may be modelled on the basis of
and with the help of mathematical apparams used in Statistical Energy Analysis (SEA)
[5] The method is especially useful to calculate the statistical approach vibroacoustical
energy flow in middle and high frequency range. With the help of a few parameters,
such'as- modal density, damping loss factor, coupling loss factor and the value of
input power, building linear equations set it is possible to describe the flow of
vibroacoustical energy in a complicated stmcture. There is also a possibility of quick
estimation of the influence of constmction method on the vibroacoustical parameters
of the whole set. In the following work an exemplary application of one of the most
frequently used software for calculating the flow of acoustic energy has been
presented- AutoSEA programme [1]. The aim of the work is practical modelling of
vibroacoustical energy flows through screw-connection of two plates and comparing
quantity results with experimental (outcome) measurements. Equivalent coupling loss
factor has been calculated for a group of mumally combined elements constimting a
construction fragment. A comparison between the measured results and the value of
coupling loss factor in linear joint (e.g. in welded one) has also been made. Using the
method of fmite elements, the influence of rubber separator thickness on the value of
the first several frequencies of free vibrations has been computed as well.
2. Physical model of plate connection
A connection of two perpendicular plates has been chosen for modelling the
flow of vibroacoustical energy in mechanical joints. Connection diagram is presented
in Figure 1. On the length of common edge the plates has been joined with anglesteel
by screws. A rubber separator (4) has been placed between the excited plate and the
anglesteel leg (3).
1143
Modelled stmcture
3. SEA model
In order to carry out the vibroacoustical analysis of the system using Statistical
Energy Analysis a model of the examined strucmre has been built. It has been
assumed that in every element of the construction only flexural waves propagate.
Every plate and the rubber layer have been modelled with just one appropriately
chosen subsystem. The anglesteel, however, has been modelled as a continuous
connection of two plates having the dimensions which correspond to the anglesteel
legs the plates themself being set at the right angle.
Fig. 2. SEA model of a system Fig- 3. Modal densities of the subsystems
1144
Using the SEA method we are able to describe the flow of the vibroacousdcal
energy in middle and high frequencies with an algebraic equation set. The exMined
system consists of five simple subsystems, of which only one is exited to vibration
with applied force. The flow of vibroacoustical energy m the model presented is
depicted with the following equation:
^ 1 tot
-^21
0
0
12
^ 2 tot
32
0
0
“'•123
^ 3 tot
-‘n43
0
0
“''134
h 4 tot
0
0
0
"'’145
0
tot
w,
0
^2 tot
0)
0
0
•
^3 tot
=
0
-^54
^4 tot
0
^ 5 tot
^5 tot
0
^ ^ 4- is a total coefficient of energy loss for every subsystem,
rj’. '"' - intemarioss factor of the subsystem,
ri.. - coupling loss factor between subsystems,
E- - the mean vibrational energy in Af frequency band in i-subsystem,
W - the input power carried into i-subsystem from outside.
To determine the elements of the coefficients matrix in eq. 1 it is necessary to
know [1,4,6] coupling loss factors (CLF) between structural subsystems and damping
In the SEA model in question transmission of the acoustic energy occurs in two
tvpes of connections between;
the plate and the beam (the point joint of the beam which is parallel to the edge
of the plate, (transfers flexural waves), . . „
two plates (linear connections and point joints transferring flexural waves).
The coupling loss factor between the plate and the beam which vibrate in the
flexural way (in the case of the point joint), is defined with following equation [1].
1.75c^
(2)
where:
c - is the speed of flexural wave,
T - transmission factor,
Q - the number of point connections,
Gj - angular frequency,
A - the surface of the plate.
1145
The speed of the flexural wave in the first plate c„ can be calculated in the
following way:
(3)
In the model under examination formula (2) defines coupling loss factors between the
anglesteel leg (2) and the beam (4) - ( factors 1,3. and tiJ or the beam (4) and the plate
(5) - rj45 and 7/54.
The flow of energy between two plates (which are connected at the right angle
and which vibrate in flexural way) is defined with the following formula in the case
of linear connection:
where:
1 - is the length of the connection.
With above formula it is possible to describe the flow of energy through correctly
made welded joints of plates or, for example, through bent plates. In the system
presented the factor determines the flow of energy between anglesteel legs (2) and
(3)
The coupling loss factor between two plates with a point joint is described by
the following formula:
iiL^
3 (o.X,
(5)
This type of connection occurs between the plate representing the anglesteel
leg and the plate (5). ^ .
After defining the value of factors matrix in the first equation it is possible to
specify the ratio of the vibroacoustical energy gathered in plate (1) and (5).
_ (n 2tof^ 3t0t~ ^ 24^ 32) 4fot^ 5tnt~ ^ 54^ 45^ ~ 34^ 43^ 5tot (5)
£5 ^21^ 32'^ 43^ 54
The damping loss factor is important parameter of every subsystem. For steel
plates used in the experiment the value of the damping loss factor have been measured
experimentally with the decay method. The results of the measurements have been
presented in figure 4. The frequency characteristic of rubber damping has been shown
in figure 5.
1146
DLF of steel [-]
frequency [Hz] frequency [Hz]
Fig. 4. Damping loss factor of steel Fig. 5. Damping loss factor of rubber
4. Experimental research
Experimental investigations have been carried out for connections made with
the use of rubber separator (elastic layer) of 50° Shore hardness. The connection was
build up of two identical, perpendicular plates connected each to another using the
anglesteel and the elastic rubber layer. These are the properties and material
parameters of individual elements:
plates:
- material constructional steel (St3);
- dimensions: 500 * 500 * 2.2 mm;
- Young modulus: 2,1 10'^ Pa.
anglesteel:
- material: constructional steel (St3);
- dimensions: L 40 40 2.2 mm;
- Young modulus: 2,1 10' ‘ Pa
elastic layer:
- material: rubber 50° Shore
- dimensions: 500 40 mm
- thickness: 2, 3, 4, 5, 6 mm
To avoid the loss of mechanical energy in the environment, during measure¬
ments the construction was suspended to the supporting frame with three weightless
strings in such way that only rigid body motions in the plane perpendicular to the plate
surface can occur. The excitation of the wide-band type with constant power spectral
density was applied in the symmetry axis of the plate (5) about 20 mm below the
upper edge (Figure 1). During the experiment the distribution of vibrating velocities
on the plate surface was obtain by non contact method using laser-vibrometry.
1147
5. FEM model
Vibration of modelled structure in low frequency has been analyzed by toe
Finite Element Method (FEM). Several FEM models, was build in order to consider
L valid thickness of the mbber layer. The mbber thickness has been from
0 mm to 6 mm. The dimensions and material parameters of the plates and the
LSesteel was constant. The structure was fixed in four comers. Calc^auons were
mfde for a division of the strucmre into 608 elements of type BRI^8. The mfluence
of the rubber thickness on the eigenfrequencies was remarkable. The results for first
15 eigenfrequencies are presented in the table 1 . In fig. was shown also the changes
of the value of natural frequencies in comparison with the natural frequencies of
strucmre without applying the mbber layer.
table 1. Namral frequencies of connection
LaDlc i. i>aLutai ai. — -
1 Natural frequencies (Hz) for various rubb
■ ■ . - - •
er layer tlrickness
0 nun
2 nun
3 mni
4 nun
5 nun
127.8
123.1
121.5
119.4
117.5
297.5
235.8
224.6
212.6
204.6
507.6
432.5
391.3
357.1
330.0
1003.2
566.9
525.7
503.7
491.9
1282.3
807.1
768.9
733.5
712.2
1900.5
1185.2
1169.4
1123.2
1070.2
2386.7
1570.2
1483.6
1376.2
1304.6
3272.3
1851.6
1634.2
1531.6
1490.8
3603.4
2303.1
2263.4
2167.8
2147.3
4263.3
2611.2
2539.5
2496.9
2463.7
4759.5
3192.7
3515
3110.1
2913.2
5438.4
3926.7
3874.6
3626.4
2937.0
7627.4
4110.1
4105.5
3639.9
3026.6
7691.6
5070.7
5137.0
3670.5
3111.2
11543.2
5547.1
5503.9
3721.9
3181.2
The decreasing of the absolute values of the natural frequences is observed according
to increasing of the thickness of the rubber layer. Beginning from die third of
°ib adons of the system the decrease of the natural frequencies is almost constant for the
"layer 2 mm or 3 mm and is continuously decreasing for rubber layer 4-6 mm (See
fig. 6).
1148
Fig. 6. Changes of natural frequencies
6. Comparison with experimental results
The equivalent coupling loss factor defining the energy flow between the plates has
been determined experimentally [31 for a model consisting of two subsystems; two plates.
The coupling loss factor in such two element model may be specified by the following
equation:
1 zast.
^2 ^1101
N, E2.0C
- 1
(7)
The quotient of plate energies E,/E, in a two-element model is relevant to the quotient of
energy in the first and fifth subsystem (E./E,) in the five element model presented m figure
0 the value of these quotient is defined with the equation (6)
In the picture we have presented the values of the equivalent coupling loss factor in
the connection. Individual points in the diagram show the results obtained experimentally.
The values received in computer simulation have been presented as a continuous diagram.
The upper curve shows the values of the coupling loss factor in the joint before the
application of the rubber layer.
In the frequency range above ca. 125 Hz we have received a very good comparison
of experimental results and computer simulation results performed with the AutoSEA
software At the frequency of about 200Hz there occurs a local minimum of the equivalent
coupling loss factor between the plates. The value of the minimum is essentially influenced
by the value of the rubber damping loss factor. The frequency (with the minimum CLF) is
strongly influenced by the peak frequency of the rubber damptng curve.
1149
o.ooo2ii -
symulation
experiment
welded plates
16' ‘ '40' ' ’lOO 250 630 1600 4000
frequency [Hz]
Fig.7 Equivalent CLF factor in the joint
7. Conclusion
A way of modelling the vibroacoustical energy flow with the help of SEA method has
been presented in the work. We have examined the screw connection of two plates, where
a rubber elastic layer has been applied, A comparison has also been made between the results
of computer simulation of the mechanical energy flow with SEA method and the experimental
results, and thus we have noticed the good correlation, especially as far as middle frequencies
The joint modification through introduction of the rubber separator has a remarkable
impact on the acoustic energy flow: , ^
- The application of the elastic layer in the Joint in question lowers the value ot
eauivalent coupling loss factor in the whole frequency range.
- the value of rubber damping factor has most significant influence on the acoustic
eneroy flow through connection in middle frequencies range:
“ The minimum value of the equivalent coupling loss factor in a joint is essentially
influenced by the rubber separator damping loss factor.
* The frequency of minimum CLF occurrence is strongly influenced by the peak
frequency of the rubber damping curve.
The increasing thickness of the rubber layer produces on decreasing natural frequencies of
the structure.
8. Bibliography:
III AutoSEA - User Guide Vibro-Acoustic Sciences Limited 1992.
121 Fahy F.J. Sound and Structural Vibration; Radiation, Transmission and Response Academic Pres
nTkirtuin J Smals N. Panuszka R. "Method of estimating the coupling loss factor for a set of
nlates" Mechanic, Technical University of Mining and Metallurgy, Cracow, 10,1991.
[^41 Lalor, N.: The evaluation of SEA Coupling Loss Factors. Proc. V School Energy Methods in
Vibroacoustics" - Supplement, Krakow-Zakopane 1996. , . „ „
[51 Lyon R., DeJong R.; Theory and Application of Statistical Energy Analysis. Butterworth-
Heinemann, Boston, 1995.
1150
Variation Analysis on Coupling Loss Factor
due to the Third Coupled Subsystem in
Statistical Energy Analysis
Hongbing Du Fook Fah Yap
School of Mechanical & Production Engineering
Nanyang Technological University
Singapore 639798
Abstract
Statistical Energy Analysis (SEA) is potentially a powerful method for
analyzing vibration problems of complex systems, especially at high frequen¬
cies. An impoitant parameter in SEA modeling is the coupling loss factor
which is usually obtained analytically based on a system with only two cou¬
pled elements. Whether the coupling loss factor obtained in the classical way
is applicable to a practical problem, which normally comprises of more than
two elements, is of importance to the success of SEA. In this paper, the varia¬
tion of coupling loss factor between two subsystems due to the presence of a
third coupled subsystem is investigated. It is shown that the degree to which
the coupling loss factor is affected depends on how strong the third subsystem
is coupled. It also depends on the distribution of the modes in the coupled sub¬
systems. This kind of effect will diminish when the damping is high, subsys¬
tems are reverberant, or ensemble-average is considered, but not for individual
cases.
1 Introduction
SEA is potentially a powerful method for analyzing vibration and acoustic problems
of complex systems, especially at high frequencies, because of the simplicity of
its equations compared to other deterministic analysis techniques. SEA models a
system in terms of interconnected subsystems. The coupling parameter between
any two subsystems is characterized by a coupling loss factor. If the coupling loss
factors and internal (damping) loss factors of all subsystems are known, the power
balance equation (e.g., see [1]) for each subsystem can be established. From this set
of equations, SEA predicts the system response (due to certain types of excitation)
in terms of the average energy of every subsystem. The energy can in turn be related
to other response quantities such as mean velocity or strain.
1151
Historically, the SEA power balance equations were initially derived from an
analysis of two coupled oscillators [2,3]. It has been shown that the energy flow
between them is directly proportional to the difference in their uncoupled modal
energies. The theory has then been extended to systems with multi-coupled sub¬
systems (e.g., [4]). Strictly this extension is only applicable if certain assumptions
are justified [4, 5]. Also the new concept of indirect coupling loss factor, which
is used to represent the energy flow proportionality between the indirectly coupled
subsystems, is also introduced.
In practice, the indirect coupling loss factors are normally ignored in SEA ap¬
plications because they are very difficult to determine analytically. Only coupling
loss factors between directly coupled substructures are considered. Some analy¬
ses [6-8] have shown that this approximation may lead to significant errors in the
predicted results if certain conditions are not met in the system. These conditions
include not only the well known requirement of weak coupling between subsystems
(e.g., see [6]), but also others, such as given by Langley that the response in each
element must be reverberant [7]; and by Kean that there should be no dominant
modes (peaks) inside the frequency-averaging band [8]. However, it is usually dif¬
ficult to know whether these conditions are satisfied for a particular system. In fact
the above mentioned conditions do not always hold for practical engineering cases.
On this point of view, the importance of a coupling loss factor for describing the
coupling between indirectly coupled subsystems are to be further examined.
A related question is whether the coupling loss factors obtained from the system
with only two subsystems can still be applied when other subsystems are present.
Generally, the coupling loss factor is sensitive to the amount of overlap between
the modes of the two coupled subsystems. When additional subsystems are cou¬
pled to the original two-subsystem model, the mode distributions of the originally
coupled two subsystems will be affected. The change of mode distributions will fur¬
ther affect the modal overlap between the coupled two subsystems and finally the
coupling loss factor between them. However, general estimation methods for cou¬
pling loss factor assume that the coupling parameters between two subsystems are
not affected much by the presence of the other subsystems. Therefore the conven¬
tional approaches of deriving coupling loss factor are mostly based on consideration
of a two-subsystem model only. One method is the wave approach, by which the
coupling loss factor used in the SEA applications are derived analytically from aver¬
aged transmission factors of waves that are transmitted through a Junction between
semi-infinite subsystems. This method only takes into account local properties at
the joints and sometimes may be inaccurate. Recent research [1,7,9-11] based
on the model with two-coupled subsystems has shown that the coupling parameter
does depend on other system properties, such as damping loss factor, etc. It can be
argued that, if there is a third coupled subsystem, the coupling parameters between
the first two subsystems will also depend on the energy flow to the third subsys¬
tem. Therefore, from a practical point of view, the coupling loss factor estimated
for two-coupled subsystems, ignoring the indirectly coupled subsystems, can only
be of approximate value.
1152
In this paper, the variation of coupling loss factor between two subsystems due
to the presence of a third coupled subsystem is studied. In the following sections,
the coupling loss factor is firstly expressed in terms of global mobility functions.
The exact solution of mobility functions is only for simple structures. However, for
general structures, it can be obtained by Finite Element Analysis (FEA) [10, 1 1].
The coupling loss factors obtained respectively in the cases with and without the
third subsystem in the model are compared for two particular system configura¬
tions, respectively. The system used in this investigation is one-dimensional simply
supported beanis coupled in series by rotational springs. By varying the spring stiff¬
ness, the strength of the coupling between the second and the third subsystems can
be changed. It is shown that the effect of the third coupled subsystem on the cou¬
pling loss factor between the first two coupled subsystems depends on how strong
the third subsystem is coupled. For each individual case, it is also shown that this
kind of effect may be positive or negative, depending on the distribution of modes
in the coupled subsystems.
2 Coupling Loss Factor by Global Modal Approach
In this section, a modal method is used to derive coupling loss factor in a sys¬
tem with any number of coupled subsystems. The result is then simplified for two
cases: (1) a three-subsystem model; (2) a two-subsystem model which is simply
substructured from the previous three-subsystem model by disconnected the third
subsystem.
For a linear system which consists of N coupled subsystems, if “rain-on-the-
roof” excitation [10] is assumed to be applied to each subsystem in turn, the corre¬
sponding response energy can be expressed as
I I drdsdu, (1)
sith.'ii/sl.Kmi Mii.bsy.'it.emj
where /?/; is the total time-averaged response energy of the subsystem i. due to
the excitation on the subsystem j, is the transfer mobility function be¬
tween the response points r and the excitation point H is the averaging range
of frequency, u.', in and S are the mass density and the power spectral density of
excitation. The input power due to the excitation is given by
I RelH(s,.i,L0)]d.sdu (2)
Q sahsyale'inj
where is the real part of the point mobility at the position
simplicity, two terms, a/, and are defined as
a,, =
i>, = ^ =
Al
m;S j
Ik
\ H [r, s,uj)\~ dr ds cl.in
n sii.bsy.'iUjvii /iubfiyslenij
lle[H{.^,s,ij)] dsd.u:
n subsyslr;mj
For
(3)
(4)
1153
The mobility function, H{r,.s,uj), is to be expressed in terms of the global modes
of the system, which can be obtained by Finite Element Analysis (FEA). By the
principle of reciprocity of the mobility function, the relation of aij = a ji always
holds regardless of the strength of coupling and the magnitude of input power if the
excitation is “rain-on-the-roof Theoretically applying the Power Injection Method
[12] we can obtain the SEA equation as
n = [77]E (5)
where H = {Hi, ila, - • • , and E = {E[, E-z, - - - , E^r}'^ . The SEA loss
factor matrix [77] is
iVl +
i.^2
(7?yV +
■niiau
niiCLiisr
bi
b-z
bN
■nua-ii
■nizCL-zN
bi
bo
bjV
77?.;Va,Yi
•m.A/a/VY
^1
Ih
b^!
= — B A“'M-
UJr
where, //; is the internal loss factor for subsystem i., v/,, is the coupling loss factor
from subsystem -I. to subsystem j, ujc is the central frequency of the averaging band
n.
A = [a,,]
From equation (6), the reciprocity principle of the coupling loss factors can be easily
seen, due to aij = aji.
1
''■bi
, M =
L ’-J
Vij n?. ; b j ni jbj/ (cu^/i / 2)
i]ji niibi ■niibi/iujc'^l'l)
Where, (77?.;6,:)/(u,v7r/2) is the averaged real part of point mobility [4] and can be
regarded as the generalized modal density of the subsystem i. Assuming weak
coupling and light damping, it approximately equals to the classical definition of
modal density [13]. Therefore, the relation given by equation (7) also reduces to the
classical reciprocity principle.
1154
2.1 Substructured two-subsystem model
Figure 1 : A general SEA model with three coupled subsystems in series
Consider a whole system with three substructures coupled in series as shown in
Figure 1 . If subsystem 3 is removed, the coupling loss factors between subsystems
1 and 2 are given in the equation
biCL22 bi(Li-2
"h '^712 — ^?2L _ ^ n?. i(fi 1 [fi22 — '^7.2(^11^22 — <'^i2^^'2l)
— 7/12 772+7/21 u-V _ ^20.21 _ _ ^2(1 1 1 _
777 i{an<^f22 “ <^^12f''2l) 7772(^110-22 — O12O21)
r bj ^to-12 1 (8)
^ J_ 777 ifli| 7772011022
62O21 ^2
7771O11O22 777.2 0 22
The approximation in the above equation is due to 011022 » 012O21 when the cou¬
pling is weak. Manipulating equation (8) with or without using the approximation
both can work out the coupling loss factors 7/12 and 7/21 as
(9)
r 61 6-,
( 777 1 I ) ( 77 72 62 ) - 7/2 - 77
07^-777-10 1 1 07^777.2022
! , ^2 . f. *^1
77 7 20'7 - 777 [Oi
"07c7772022 07^-777-1 011
(10)
The equations are true regardless of the strength of the coupling. It can be seen that
7/12 and 7/21 depend on the values of the three terms 777.,+/, 7// and bij{u:^ni-,au). The
first two are the generalized modal density and the internal loss factor, or in combi¬
nation equivalent to modal overlap factor. The third one, by noting the definitions
of 6; and an, is the ratio of input power to response energy for the directly excited
subsystem, i.e., the total loss factor of subsystem i. From equation (8), this term
can be approximately expressed as
Total loss factor
of subsystem i
b;
miLOcCLii
m + 5I'+-.7
(11)
1155
In the classical wave approach, where semi-infinite subsystems are assumed, the
total internal loss factors becomes
■n total, I = m + Vn and l]iot.al,2 = 112 + vTi (^2)
where is the classical coupling loss factor. Substituting equation (12) into equa¬
tions (9) and (10), i]ij reduces to the classical iiff, which only depends on the local
properties at the joints rather than other properties of the system, such as damping.
If the subsystem modal parameters are used to evaluate the term, bil{ujcmiaii), then
the total loss factor is Just the internal loss factor of the subsystem and the cou¬
pling loss factor is equal to zero. This is reasonable because using the uncoupled
modal parameters instead of the coupled modal parameters is actually equivalent to
removing the coupling between two subsystems.
However, for finite system where the assumption of semi-infinity is not justified,
there will be no immediate simplication for equations (9) and (10). Numerically,
FEA can be employed to obtain the global modes and then the coupling loss factor
can be calculated [10, 1 1].
2.2 Full three-subsystem model
Instead of substructuring, consider the three-subsystem model as a whole system,
shown in figure 1. Now the order of equation (6) is reduced to 3. With the global
modal parameters obtained from FEA, the coupling loss factors can be directly
evaluated. However, when the coupling between subsystems is weak, the order-
reduced equation (6) is still able to be simplified. Matrix A may be alternatively
expressed in the form of
■ a, L
0
0 ‘
■ 0
«L2
0
■ 0
0
«13 '
A -
0
(t-ll
0
-f
a\2
0
d.23
+
0
0
0
0
0
«33 .
0
((•23
0 .
. (l.[3
0
0
where the terms on the right side are sequentially defined as Ai, A2 and A3 . Under
the assumption of weak coupling, the non-zero entries in Ai, A2 and A3 will be of
the order O(t^), (9(e^) and respectively [14], The inverse of matrix A may
be approximately written as
= Ai”^ — Ai“^ A2 Ai”^ — Ai"^ A3 Ai ^ 4- Ai ^A2Ai ^A2Ai
+ •••
(14)
As an approximation, substituting only the first three terms in equation (14) into
equation (6) gives
[tj] ^ l/uv-
/>■
bo f t’ 12
rn I f( 1 1 a 2 2
b3(l-\3
ni[a\ KM. 3
■ni2a 1 1 (1-22
b2
1112(1.22
63^23
1112(1.22^(33
bi(t.\3
in3(l\\(l.33
bid. 13
in.3Ct.2'i((33
63 ~
1((3((33
(15)
1156
Generally, under the condition of weak coupling, the indirect coupling loss factors,
■ihs and 7731, are much smaller than the direct coupling loss factors and the internal
loss factors [5,7, 15]. The diagonal elements in [rj] can therefore be approximated to
the sum of internal loss factor and direct loss factor. It can be shown from equation
(15) that equation (11) remains valid for three coupled subsystems. But in the three-
subsystem case, the term, b,|{uJcm.^au), is to be evaluated by using the global modes
of the three-subsystem model.
3 Numerical Examples and Variation Analysis
In this section, two examples with different configurations are used to show the vari¬
ation of coupling loss factor due to the presence of a third coupled subsystem. The
coupling loss factor of the two-subsystem model is evaluated by using equations (9)
and (10). For the three-subsystem model, equations (6) and (15) are used. It can be
shown that both equations (6) and (15) give the same results as the couplings are
weak.
3.1 Structural details and SEA model
Ki
beam I (TJT] -
(a) ^ - Zi.
beam 1 rnn beam 2 (TTH ^
(b) ^ ^ - A A -
Figure 2; A structural model comprising of three beams
To begin example calculations, consider initially a two-subsystem model (figure 2(a))
which is two thin beams coupled through a rotational spring. The group of flexural
vibration modes of each beam are taken as a SEA subsystem. The spring provides
weak coupling between them where only rotational moment is transmitted. When
beam 3 is connected at the free end of beam 2 to the original two-beam model,
a three-subsystem model is formed (figure 2(b)). The specifications for the three
beams are given in table 1. The spring stiffness, A'2, is adjustable in order to look
into the significance of the effect of the third subsystem. There are two cases where
the length of beam 2 is: (i) L2 - 1.0?72; (ii) L2 = l-lm. The spring constants at
the joints are chosen to be weak enough to ensure that:(a) the coupling loss factor
is much smaller than the internal loss factor; (b) the indirect coupling loss factor is
much smaller than the direct coupling loss factor.
In the global modal approach (see section 2), the modes of two-subsystem model
and three-subsystem model are obtained from FEA. In numerical simulation, the
1157
Table 1 : The specifications of the three beams
BEAM
1
2
3
length (m)
2.0
1.0 & 1.1
0.7
width (mm)
4
Thickness (mm)
2
Density (Kg/m^)
7890
Young’s Modulus (N/m'^)
196E+9
Poisson Ratio
0.29
spring constant, A*i (Nm/rad)
1.0
central frequency is 200Hz and the averaging band is selected as 100 ~ 300Hz. In
order to take into account the contribution from the modes out of the band, all the
modes up to 500Hz are extracted for evaluating the mobility functions in averaging.
The modal loss factor is assumed to be the same for each modes used in averaging.
This means that the internal loss factor is the same for each subsystem and is equal
to the modal loss factor [11]. The results given are plotted against the modal loss
factor in order to show the damping effect at the same time.
3.2 Results and discussion
Figure 3 shows the identified coupling loss factor 7/12 for the case (i) {Lo = 1.0777)
with different stiffness of /v'2. The case of /v'2 = 0 means that the third subsystem
Figure 3: 7/12 is negatively affected in three-subsystem model
is not present. It can be seen that the coupling loss factor 7^12 is decreased in the low
1158
range of damping while the strength of the coupling between subsystem 2 and 3 is
increased. The stronger the coupling, the more ijn is decreased.
On the other hand, for the case (ii) where L2 = l.lm, the different results are
shown in figure 4 where the presence of the third subsystem would mainly increase
i]i2 in the low range of damping. The increasing magnitude is also dependent on the
strength of coupling between subsystem 2 and 3. The explanation for the different
variation trends of ?]i2 due to the third coupled subsystem between figure 3 and 4
will be given later.
Figure 4; 771 ■; is positively affected in three-subsystem model
From figure 3 and 4, the effect of damping on the coupling loss factor can also be
observed. In the low damping region, increasing damping would increase coupling
loss factor. After a certain turnover point, increasing damping would make the
coupling loss factor decrease and finally 7712 becomes convergent to a value. This
agrees with the conclusions drawn in [10, 11]. It is shown that, even though the
length of beam 2 has a slight difference in figure 3 and 4, the converged values are
still very close. Thus, the converged value seems not to depend on the variation of
coupling strength at A'2 and the structural details, although, with the third subsystem
existing in the system, the convergent speed is faster. Therefore, it is reasonable to
believe that the converged coupling loss factor at sufficiently high damping only
depends on the property of the joint rather than other system properties. This joint-
dependent property of coupling loss factor in the high range of damping accords
with the assumption in the wave approach. Here, it is convenient to define the
convergent region in the figure 3 and 4 as the “joint-dependent zone”.
However, before the “joint-dependent zone”, coupling loss factor seems very
sensitive to the variation of damping loss factor as well as the strength of coupling
between subsystem 2 and 3. It is because in the low damping region the system
1159
modal properties have been playing a major role in determining coupling loss fac¬
tor [10, 11]. In general, the coupling loss factor represents the ability of energy
transmitted between subsystems. It depends not only on the physical strength of
the coupling (e.g., spring stiffness in the examples), but also on the amount of over¬
lap between the modes of two connected subsystems. The higher modal overlap
between the modes of two connected subsystem, the more energy is transmitted
between the subsystems. As a result, the coupling loss factor will be higher even
though the physical strength at the joint is unchanged. If the modes in one subsys¬
tem are distributed exactly the same as those in one another(for instance, two exactly
same structures are coupled together), the coupling loss factor would be varied to
the maximum, and vice versa. Therefore, this region could be likely defined as
“modal-sensitive zone”.
In the “modal-sensitive-zone”, the dependence of coupling loss factor on the
amount of overlap between the modes of two connected subsystems has been clearly
shown in figures 3 and 4. For the case (i) illustrated in figure 3, the length of beam 2
is half of beam 1. Due to the characteristic of mode distribution in beam structure,
the amount of overlap between the modes of subsystem 1 and 2 is more than that in
the case (ii) shown in figure 4, where beam 1 is 2 meters and beam 2 is 1.1 meters.
Therefore, case (i) has higher coupling loss factor than case (ii) in “modal-sensitive
zone”. When the third beam is coupled, the induced variation of coupling loss factor
depends on how the amount of overlap between the modes of subsystems 1 and 2 is
affected. It can be increased or decreased and thus the coupling loss factor between
subsystems 1 and 2 can also be increased or decreased due to the third coupled
subsystem. For example, the amount of such overlap in case (i) is decreased after
the third subsystem is coupled. As a result, the coupling loss factor, 7712, becomes
decreased.
The above discussed variability of coupling loss factor due to the third coupled
subsystem has been shown for individual cases. On the other hand, if an ensemble
of similar structures are considered, this sensitivity may be reduced (as it is some¬
times positive or negative depending on each special situation). However, such a
variability obtained from two typical examples is nevertheless very useful when
one individual case is studied in SEA or SEA-like problems. The ignorance of such
effect of the other coupled subsystems on the coupling loss factor may become one
of the possible error sources causing SEA failure.
4 Conclusions
The variation of coupling loss factor due to the third coupled subsystem is stud¬
ied in this paper. The effect of a third coupled subsystem on the coupling loss
factor between the first two coupled subsystems depends on how strong the third
subsystem is coupled. Roughly, along with the damping in the subsystems, “joint-
dependent zone” and “modal-sensitive zone” are defined according to the different
variation properties of coupling loss factor. In the “modal-sensitive zone”, the ef-
1160
feet of a third coupled subsystem on the coupling loss factor could be positive or
negative. It depends on how the amount of overlap between the modes of two con¬
nected subsystems is affected. This “modal-sensitive” effect may be averaged out
for an ensemble of structures, but it is important when SEA is applied to individual
cases. In the “joint-dependent zone”, the coupling loss factor is insensitive to the
strength of the coupling between the second and third subsystems. Since the two
different zones are allocated according to the system damping (which is equivalent
to modal overlap factor when the central frequency and modal density are fixed),
it shows the importance of reverberance in subsystems when the classical SEA is
applied [7,8]. How to take into account the effect of the other coupled subsystems
in evaluating coupling loss factor, especially when the system damping is low and
when an individual case is considered, definitely needs to be further investigated.
References
[ 1 ] Richard H. Lyon and Richard G. DeJong. Theory and Application of Statistical
Energy Analysis. Butterworth-Heinemann, second edition, 1995.
[2] Richard H. Lyon and G. Maidanik. Power flow between linearly coupled os¬
cillators. Journal of the Acoustic Society of America, 34:623-639, 1962.
[3] Eric E. Ungar. Statistical energy analysis of vibrating systems. Transactions
of the ASME, Journal of Engineering for Industry, pages 626-632, November
1967.
[4] F. J. Fahy. Statistical energy analysis. In R. G. White and J. G. Walker, editors.
Noise and Vibration, chapter 7, pages 165-186. Chichester, Ellis Horwood,
1982.
[5] J. M. Cuschieri and J. C. Sun. Use of statistical energy analysis for rotating
machinery, part II; Coupling loss factors between indirectly coupled substruc¬
tures. Journal of Sound and Vibration, 170(2): 191-201, 1994.
[6] P. W. Smith. Statistical models of coupled dynamical systems and the transi¬
tion from weak to strong coupling. Journal of the Acoustic Society of America,
65:695-698, 1979.
[7] S. Finnveden. Ensemble averaged vibration energy flows in a three-element
structure. Journal of Sound and Vibration, 187(3);495-529, 1995.
[8] A. J. Keane. A note on modal summations and averaging methods as ap¬
plied to statistical energy analysis (SEA). Journal of Sound and Vibration,
164(1); 143-156, 1993.
[9] B. R. Mace. The statistical energy analysis of two continuous one-dimensional
subsystems. Journal of Sound and Vibration, 166(3):429-461, 1993.
1161
[ 10] Hongbing Du and Fook Fah Yap. A study of damping effects on coupling loss
factor used in statistical energy analysis. In Proceedings of the Fourth Inter¬
national Congress on Sound and Vibration, pages 265-272, St. Petersburge,
Russia, June 1996.
[11] Fook Fah Yap and J. Woodhouse. Investigation of damping effects on statis¬
tical energy analysis of coupled structures. Journal of Sound and Vibration,
197(3):35I-371, 1996.
[12] D. A. Bies and S. Hamid. In situ determination of loss and coupling loss
factors by the power injection method. Journal of Sound and Vibration,
70(2): 187-204, 1980.
[13] L. Cremer, M. Heckl, et al. Structure-Borne Sound: Structural Vibrations and
Sound Radiation at Audio Frequencies. Springer- Verlag, second edition, 1987.
[14] R. S. Langley. A derivation of the coupling loss factors used in statistical
energy analysis. Journal of Sound and Vibration, 141(2):207-219, 1990.
[15] J. C. Sun, C. Wang, et ai. Power flow between three series coupled oscillators.
Journal of Sound and Vibration, 1 89(2) :2 15-229, 1996.
1162
THE EFFECT OF CURVATURE UPON VIBRATIONAL
POWER TRANSMISSION IN BEAMS
SJ. Walsh(l) and R.G.White(2)
(1) Department of Aeronautical and Automotive
Engineering and Transport Studies
Loughborough University
(2) Department of Aeronautics and Astronautics
University of Southampton
ABSTRACT
Previous research into structural vibration transmission paths
has shown that it is possible to predict vibrational power
transmission in simple beam and plate structures. However, in
many practical structures transmission paths are composed of more
complex curved elements; therefore, there is a need to extend
vibrational power transmission analyses to this class of structure.
In this paper, expressions are derived which describe the vibrational
power transmission due to flexural, extensional and shear types of
travelling wave in a curved beam which has a constant radius of
curvature. By assuming sinusoidal wave motion, expressions are
developed which relate the time-averaged power transmission to
the travelling wave amplitudes. The results of numerical studies
are presented which show the effect upon power transmission
along a curved beam of: (i) the degree of curvature; and (ii) various
simplifying assumptions made about the beam deformation.
1. INTRODUCTION
Previous research into structural transmission paths has
shown it is possible to predict vibrational power transmission in
simple beam and plate structures. More recently, transmission
through pipes with bends, branches and discontinuities has been
studied, which has led to useful design rules concerning the
position and size of pipe supports for minimum power
transmission[l]. However, in many practical structures
transmission paths are composed of more complex curved
elements. Therefore, there is a need to extend power transmission
analyses to this class of structure.
1163
Wave motion in a curved beam with a constant radius of
curvature has been considered by Love [2] who assumed that the
centre-line remains unextended during flexural motion, whilst
flexural behaviour is ignored when considering extensional
motion. Using these assumptions the vibrational behaviour of
complete or incomplete rings has been considered by many
researchers who are interested in the low frequency behaviour of
arches and reinforcing rings. In reference [2] Love also presented
equations for thin shells which include the effects of extension of
the mid-surface during bending motion. Soedel [3] reduced these
equations and made them applicable to a curved beam of constant
radius of curvature. In an alternative approach Graff [4] derived
these equations from first principles and also constructed frequency
verses wavenumber and wavespeed versus wavenumber graphs.
Philipson [5] derived a set of equations of motion which included
extension of the central line in the flexural wave motion, and also
rotary inertia effects. In a development analogous to that of
Timoshenko for straight beams, Morley [6] introduced a correction
for radial shear when considering the vibration of curved beams.
Graff later presented frequency versus wave number and wave
speed versus wave number data for wave motion in a curved beam,
when higher order effects are included [7].
In this paper, expressions for vibrational power transmission
in a curved beam are derived from first principles. In the next
section two sets of governing equations for wave motion in a
curved beam are presented both of which include coupled
extensional-flexural motion. The first set is based upon a reduction
of Love's thin shell equations mentioned above. The second set is
based upon a reduction of Fliigge’s thin shell equations [8]. In
section three, the expressions for stresses and displacements
presented in section two are used to derive formulae for vibrational
power transmission in terms of centre-line displacements. By
assuming sinusoidal wave motion, expressions are developed
which relate the time-averaged power transmission to the
extensional and flexural travelling wave amplitudes. In section
four, corrections for rotary inertia and shear deformation are
introduced. The results of numerical studies of these expressions
are presented which show the effect upon wave motion and power
transmission of (i) the degree of curvature, and (ii) the various
simplifying assumptions made about the beam deformation.
1164
2. WAVE MOTION IN CURVED BEAMS
In this section the governing relations between displacements,
strains, stresses and force resultants in a curved beam are presented.
The centre-line of the beam lies in a plane and forms a constant
radius of curvature. The cross-section of the beam is uniform and
symmetrical about the plane and it is assumed that there is no
motion perpendicular to the plane. It is also assumed that the beam
material is linearly elastic, homogeneous, isotropic and continuous.
Consider a portion of the curved beam, as shown in Figure 1.
The circumferential coordinate measured around the centre-line is
s, while the outward pointing normal coordinate from the centre¬
line is z, and the general radial coordinate is r. A complete list of
notation is given in the appendix. For small displacements of thin
beams the assumptions, known as "Love's first approximation" in
classical shell theory, can be made [8]. This imposes the following
linear relationships between the tangential and radial
displacements of a material point and components of displacement
at the undeformed centre-line:
U (r, s, t) = u (R, s, t) + z (|) (s, t) (1)
W (r, s, t) = w (R, s, t) (2)
where u and w are the components of displacement at the centre¬
line in the tangential and radial directions, respectively, (j) is the
rotation of the normal to the centre-line during deformation:
(|>
3w
/ angle of'X { rotational displacement^
Vcurvature/ \ of straight beam )
(3)
and W is independent of z and is completely defined by the centre¬
line component w.
Circumferential strain consists of both an extensional strain
and bending strain component. Expressions for these are listed in
table 1. The strain-displacement expressions of the Love and Flugge
based equations are identical. However, in the total circumferential
strain of the Love based theory, the term in the denominator
has been neglected with respect to unity. Assuming the material to
be linearly elastic, the circumferential stress-strain relationship
is given by Hooke's Law, whilst the shear strain, Ysr/ and shear
stress, Osr/ are assumed to be zero. Assuming the material
1165
to be homogeneous and isotropic, the material properties E, G and v
can be treated as constants. Thus, by integrating the stresses over
the beam thickness, force and moment resultants can be obtained,
which are listed in table 2. The adopted sign convention is shown in
Figure 2.
Equations of motion for a curved beam are presented in [4].
These equations are derived in terms of the radian parameter 0. By
applying the substitution, s - RQ, the equations of motion can be
expressed in terms of the circumferential length, s. These equations
are listed in [9] along with the Fliigge based equations of motion
which have been obtained by a reduction of the equations of
motion for a circular cylindrical shell presented in [8]. An harmonic
solution of the equations of motion can be obtained by assuming
that extensional and flexural sinusoidal waves propagate in the
circumferential direction. The harmonic form of the equations of
motion are also listed in [9].
3. VIBRATIONAL POWER TRANSMISSION IN CURVED BEAMS
In this section the expressions for displacements and stresses
presented in section two are used to derive the structural intensity
and power transmission due to flexural and extensional travelling
waves in a curved beam. The structural intensity expressions are
formulated in terms of displacements at the centre-line. By
assuming sinusoidal wave motion, expressions are developed
which relate the time-averaged power transmission to the flexural
and extensional travelling wave amplitudes.
Structural Intensity in the circumferential direction of a curved
beam is given by [10]:
Is =
au
at
f intensity due to
I circumferential stress
aw
at
f intensity due to
radial shear stress
(4)
By integrating across the beam thickness power transmission per
unit length in the circumferential direction is obtained:
h/2
Ps= J Isdz (5)
-h/2
1166
Substituting the Love based circumferential stress-strain relation
and strain-displacement expression into equation (5) the power
transmission due to circumferential stress is obtained. (A full
derivation is given in [9].) By analogy to power transmission in a
straight beam [1] this can be expressed in terms of an extensional
component, Pe, and a bending moment component, Pbm* Although
the transverse shear stress Ogr is negligible under Love's first
approximation, the power transmission due to transverse shear
stress can be evaluated from the non-vanishing shear force, Q,
because the radial displacement W does not vary across the beam
thickness. Again, by analogy to power transmission in a straight
beam [1] this is expressed as a shear force component. Thus, the total
power transmission in the circumferential direction is given by the
sum of the extensional, bending moment and shear force
components. These equations are listed in table 3 along with Fliigge
based power transmission equations which are also listed in table 3.
Substituting harmonic wave expressions into the Love and Fliigge
based power transmission equations gives expressions for the power
transmission in the circumferential direction in terms of travelling
wave amplitudes A and B. For sinusoidal wave motion it is useful
to develop time-averaged power transmission defined by [1]:
T/2
<Ps>
-- f
t“T J
Ps (s,t) dt
- T/2
(6)
where T is the period of the signal. Time averaged Love and Fliigge
based power transmission equations are given in table 4.
4. THE EFFECT OF ROTARY INERTIA AND SHEAR
DEFORMATION
It is known that shear deformation and rotary inertia effects
become significant for straight beams as the wave length approaches
the same size as the thickness of the beam, and for cylindrical shells
as the shell radius decreases [8]. Thus, the objective in this section is
to establish more complete equations for power transmission in a
curved beam and to show under what conditions these specialise to
the simple bending equations presented in section three.
Rotary inertia effects are included by considering each element
of the beam to have rotary inertia in addition to translational
inertia. Equations of motion for a curved beam which include the
1167
effect of rotary inertia are presented in [7]. These equations are
listed in [9] in terms of the circumferential distance parameter, s.
Equations for vibrational power transmission can be derived in the
same marmer as described in section three. These equations are
listed in tables 3 and 4 where it can be seen that the extensional and
bending moment components when including rotary inertia effects
are identical to the corresponding Fliigge based expressions.
However, the shear force component now contains an additional
rotary inertia term.
If shear deformation is included then Kirchoff s hypothesis is
no longer valid, and the rotation of the normal to the centre-line
during bending, (|), is no longer defined by equation (3) but is now
another independent variable related to the shear angle, y.
However, unlike simple bending theory, where the transverse
shear strain, Ysr^ is negligible, the transverse shear strain is now
related to the shear angle, y which is expressed in terms of
displacements u, w and (j). The circumferential force, bending
moment, and shear force obtained from [8] are given in table 2. A set
of equations of motion for a curved beam which includes the effect
of shear deformation is presented in [9]. Power transmission
equations in the circumferential direction can be obtained in a
manner analogous to that used for Love and Fliigge based theories.
As before, the power transmission due to circumferential stress
can be identified as consisting of extensional and bending moment
components. The contribution to the power transmission from the
transverse shear stress is obtained from the product of the shear
force resultant and the radial velocity which gives the shear force
component of power transmission.
5. NUMERICAL STUDY
For a given real wavenumber, k, the harmonic equations of
motion were solved to find the corresponding circular frequency, co,
and complex wave amplitude ratio. The simulated beam was
chosen to have the physical dimensions and material properties of
typical mild steel beams used for laboratory experiments. Four
different radii of curvature were investigated, which were
represented in terms of the non-dimensional thickness to radius of
curvature ratio, h/R. These ratios were ^/lO/ 100/ 1000 and
^/lOOOO-
Using the Love equations of motion. Figure 3 shows the
relationship between wave number and frequency for a beam with a
1168
thickness to radius of curvature ratio of ^/lo- The frequency range
is represented in terms of the non-dimensional frequency
parameter Q = ®^/co, where Cq is the phase velocity of extensional
waves in a straight bar and the wave number range is represented
in terms of the non-dimensional wave number, kR. It can be seen
that two types of elastic wave exist: one involving predominantly
flexural motion; the other predominantly extensional motion.
However, for wave numbers less than kR = 1, the predominantly
flexural wave exhibits greater extensional than flexural motion.
Solution of the shear deformation equations of motion for a
curved beam shows that three types of elastic wave exist. These are
the predominantly flexural and predominantly extensional waves
of simple bending theory and additionally a predominantly
rotational wave related to the shear angle. The relationship
between wave number and frequency for these three wave types is
shown in Figure 4.
A numerical investigation of the power transmission
equations was undertaken using simulated beams with the same
dimensions and material properties as those used in the previous
study of wave motion. Figure 5 shows the relationship between
transmitted power ratio and frequency. For the predominantly
flexural wave the time-averaged transmitted power ratio is
calculated by dividing the time-averaged power transmitted along a
curved beam by a predominantly flexural wave by the time-
averaged power transmitted by a pure flexural wave travelling in a
straight Euler-Bernoulli beam. i.e. the ratio (<Pe>t + <Fbm>t +
<Psf>t)/EIcokf3Af . For the predominantly extensional wave the
transmitted power ratio is calculated by dividing the time-averaged
power transmitted along a curved beam by a predominantly
extensional wave by the time-averaged power transmitted by a pure
extensional wave in a straight rod. i.e. the ratio (<Pe>t + <Pbn>t +
<Psf >t)/EScokex P^ex
6. SUMMARY AND CONCLUSIONS
In this paper, starting from first principles, expressions for
vibrational power transmission in a curved beam have been
derived using four different theories. Love’s generalised shell
equations include extension of the centre-line during bending
motion were the first set of equations considered. Fiiigge's
equations also include centre-line extensions and were the second
set of equations used. Corrections for rotary inertia and shear
deformation produced the third and fourth sets of governing
1169
equations, respectively. By letting the radius of curvature, R, tend
to infinity these equations reduce to the corresponding straight
beam expressions presented in [1].
Using the governing equations for each theory, expressions
were then developed which related time-averaged power
transmission to the amplitudes of the extensional, flexural and
rotational displacements. For each theory the effects of curvature
upon the resulting wave motion and power transmission were
then investigated using beams with different degrees of curvature.
From the results of this study it can be seen that vibrational power
transmission in curved beams can be classified into three different
frequency regions:
(i) below the ring frequency, Q = 1, curvature effects are
important;
(ii) above the ring frequency but below the shear wave cut-on
frequency, = 1 the curved beam behaves essentially as a
straight beam;
(iii) above the shear wave cut-on frequency, higher order effects
are important.
ACKNOWLEDGEMENT
The analytical work presented in this paper was carried out while
both authors were at the Institute of Sound and Vibration Research,
University of Southampton. The financial support of the Marine
Technology Directorate Limited and the Science and Engineering
Research Council is gratefully acknowledged.
REFERENCES
1. J.L. HORNER 1990 PhD thesis, University of Southampton
Vibrational power transmission through beam like structures.
2. A.E.H. LOVE 1940 Dover, Nezv-York. A treatise on the
mathematical theory of elasticity.
3. W. SOEDEL 1985 Dekker, New York. Vibrations of shells and
plates
4. K.F. GRAFF 1975 Clarenden Press, Oxford. Wave motion in
elastic solids
5. L.L. PHILIPSON 1956 Journal of Applied Mechanics 23, 364-
366. On the role of extension in the flexural vibrations of rings.
6. L.S.D. MORLEY 1961 Quarterly Journal of Mechanics and
Applied Mathematics 14, (2), 155-172. Elastic waves in a
naturally curved rod.
1170
7. K.F. GRAFF 1970 IEEE Transactions on Sonics and Ultrasonics,
SU-17 (1), 1-6. Elastic wave propagation in a curved sonic
transmission line.
8. A.W. LEISSA 1977 NASA SP-288, Washington DC Vibrations
of shells.
9. S.J. WALSH 1996 PhD thesis, University of Southampton.
Vibrational power transmission in curved and stiffened
structures.
10. A.J. ROMANO, P.B. ABRAHAM, E.G. WILLIAMS 1990 Journal
of the Acoustical Society of America 87. A Poynting vector
formulation for thin shells and plates, its application to
structural intensity analysis and source localization. Part I:
Theory.
APPENDIX: NOTATION
A flexural wave amplitude
Af amplitude of a purely flexural wave
B extensional wave amplitude
Bex amplitude of a purely extensional wave
C rotation wave amplitude
E Young’s modulus
G shear modulus
I second moment of area of cross-section of beam
Ig structural intensity in circumferential direction
K radius of gyration
M bending moment on cross-section of beam
N circumferential force on cross-section of beam
P transmitted power
Q shear force on cross-section of beam
R radius of curvature
S cross-sectional area of beam
T period of wave
LF displacement in circumferential direction
W displacement in radial direction
Co wavespeed of extensional waves in a straight bar
Cs wavespeed of shear waves in a straight bar
ds length of elemental slice of curved beam
eg total circumferential strain
h thickness of beam
k wavenumber
kex wavenumber of a purely extensional wave
kf wavenumber of a purely flexural wave
r coordinate in radial direction
1171
s coordinate in circumferential direction
u displacement at centre-line in circumferential direction
w displacement at centre-line in radial direction
z coordinate of outward pointing normal
Q. non-dimensional frequency
Ps bending strain
Y shear angle
Ysr transverse shear strain
Ej. radial strain
£g circumferential strain
0A phase angle of flexural wave
0C phase angle of rotational wave
K Timoshenko shear coefficient
?iex wave length of extensional waves in a straight bar
V Poisson's ratio
Oj. radial stress
Gg circumferential stress
0SJ transverse shear stress
d change in slope of normal to centre-line
CO radian frequency
W(r,s,t)
1172
Figure 2: Sign convention and force resultants on an elemental
slice of curved beam
Figure 3: Wave number v. frequency relationship for a curved
beam predicted using Love theory
Transmitted power ratio (Curved beam)/(Slraight beam) ^ Non-dimensional wavenumber
Table 1: Displacement, strain-displacement and stress-strain equations for a curved beam.
1175
Table 2: Force resultants for a curved beam.
1176
Table 3: Power transmission for a curved beam.
1177
Table 4: Time-averaged power transmission by a single harmonic wave
1178
A Parameter-based Statistical Energy
Method for Mid-frequency
Vibration Transmission Analysis
Sungbae Choi, Graduate Student Research Assistant
Matthew P. Castanier, Assistant Research Scientist
Christophe Pierre, Associate Professor
Department of Mechanical Engineering and Applied Mechanics
The University of Michigan
Ann Arbor, MI 48109-2125, USA
Abstract
Vibration transmission between two multi-mode substructures con¬
nected by a spring is investigated. A classical Statistical Energy
Analysis (SEA) approach is reviewed, and it is seen that some typ¬
ical assumptions which are valid at high frequencies lose accuracy
in the mid-frequency range. One assumption considered here is
that of an identical probability density function (pdf) for each reso¬
nant frequency. This study proposes a Parameter-based Statistical
Energy Method (PSEM) which considers individual modal informa¬
tion. The results of PSEM have good agreement with those of a
Monte Carlo technique for an example system.
Nomenclature
E[ ]
expected value
power transmitted between substructure i and k
IIfc(u;) .
total power transmitted to substructure k
Pi
power input to substructure i
Vi,
coupling loss factor (CLF)
fh
modal driving force for mode j of substructure 1
UJ
frequency [rad/sec]
l.l.-.lj.Ol
subscripts for decoupled Bar 1
2,2r,2„02
subscripts for decoupled Bar 2
El, ^2
blocked energy
Eoi , Eo2
Young’s modulus
Pl^P2
density
mi, m2
mass per unit length
Mu M2
total mass
Cl,C2
viscous damping ratio
Ai, A2
cross-sectional area
0
rH
0
nominal length
1179
Ti,T2
disordered length
£1,^2
ratio of disorder to nominal length
k
coupling stiffness
coupling ratio
Xi,X2
position coordinate
ai,a2
point- coupling connection position
Wi, W2
deflection
modal amplitude
mode shape function
NuN2
number of resonant frequencies
resonant frequencies
lower limit of resonant frequencies
upper limit of resonant frequencies
(Tl, (72
standard deviation of disorder
1 Introduction
Vibration transmission analysis between connected substructures in the
mid-frequency range is often a daunting prospect. Since the analysis at
high frequencies requires greater model discretization, the size and com¬
putational cost of a full structure model (e.g., a Finite Element model)
can become prohibitive. Also, as the wavelengths approach the scale of
the structural variations, uncertainties (tolerances, defects, etc.) can sig¬
nificantly affect the dynamics of the structure. Starting at what may be
called the mid-frequency range, deterministic models fail to predict the
response of a representative structure with uncertainties.
Therefore, in the mid-frequency range, a statistical analysis of vibra¬
tion transmission may be more appropriate. This approach is taken in
the procedure known as Statistical Energy Analysis (SEA) [1]. In SEA,
a structure is divided into coupled substructures. It is assumed that each
substructure exhibits strong modal overlap which makes it difficult to dis¬
tinguish individual resonances. Therefore, the resonant frequencies are
treated as random variables, each with an identical, uniform probability
density function (pdf) in the frequency range of interest. This assumption
greatly simplifies the evaluation of the expected value of transmitted vi¬
bration energy. A simple linear relation of vibration transmission between
each pair of substructures is retrieved. The power transmitted is propor¬
tional to the difference in the average modal energies of the substructures.
This relation is analogous to Fourier’s law of heat transfer [1-4].
In the low- to mid-frequency region, the modal responses are not strongly
overlapped. In this case, two typical SEA assumptions are less accurate: an
identical pdf for all resonant frequencies, and identical (ensemble-averaged)
1180
values of the associated mode shape functions at connection positions. In
this paper, these two assumptions are relaxed. A distinct uniform pdf
is applied for each resonant frequency, and a piecewise evaluation of the
transmitted power is performed. This is called a Parameter-based Sta¬
tistical Energy Method (PSEM) because it considers the statistical char¬
acteristics of individual system parameters. This solution can accurately
capture peaks of transmitted power while maintaining the SEA advantage
of efficiency.
This paper is organized as follows. In section 2, we briefly review SEA
along with the associated assumptions and limitations. In section 3, the
power transmitted between two spring- coupled multi-mode substructures
is investigated by applying several SEA assumptions. A Monte Carlo solu¬
tion is used for comparison. In section 4, the PSEM approach is presented
and the results are shown. Finally, section 5 draws conclusions from this
study.
2 Overview of SEA
In Statistical Energy Analysis, the primary variable is the time-averaged
total energy of each substructure. This is called the blocked energy, where
blocked means an assumed coupling condition. The assumed coupling con¬
dition may be the actual coupling, a clamped condition at the substructure
junctions, or a decoupled condition [2, 5].
In order to predict the average power transmitted between two directly-
coupled substructures, a few simplifying assumptions are applied. Some of
the essential SEA assumptions are summarized by Hodges and Woodhouse
in Ref. [3];
• Modal incoherence: the responses of two different modal coordinates
are uncorrelated over a long time interval
• Equipartition of modal energy: all modes within the system have the
same kinetic energy
The above conditions make it possible to treat all modal responses as sta¬
tistically identical. The first assumption implies a broad band, distributed
driving force (often called ”rain on the roof’) which leads to uncorrelated
modal driving forces. The second assumption implies that the substruc¬
tures have strong modal overlap, or that the parameter uncertainties are
sufficiently large that the modes are equally excited in an ensemble average
sense. Thus, the resonant frequencies are treated as random variables with
identical, uniform probability density functions (pdfs) for the frequency
range of interest.
The SEA relation for the expected value of power transmitted from
1181
substructure i to substructure k, may be expressed as
E[nijfc(cj)] = 0)7}, ^Ni{^ (1)
where uj is the frequency, 77^^ is the coupling loss factor, Ei is the blocked
energy of substructure z, and Ni is the number of participating modes of
substructure i for the frequency range of interest. The power dissipated by
substructure i is expressed as
= u:riiEi (2)
where r]i is the damping loss factor. Using Eqs.(l) and (2), the equation
of power balance for substructure i at steady state [1, 2, 5, 6] is
N N
Pi = E[Ilij] + UJTJiEi = UJ (3)
i=i
j¥i
where Pi is the power input to substructure i from external sources. Note
that the first term on the right-hand side is the power transmitted through
direct coupling between substructures.
3 Vibration Transmission in a Two-Bar
System
The longitudinal vibration of the structure shown in Fig.l is considered
in this study. The structure consists of two uniform bars with viscous
damping which are coupled by a linear spring of stiffness k. The spring is
connected at intermediate points on the bars, Xi = ai and X2 = 0,2. Bar
i has nominal length Loi. A parameter uncertainty may be introduced by
allowing the length to vary by a small random factor e^, which is referred
to as disorder. The length of a disordered bar is Li = Zrox(l + £:)• The ratio
of the connection position to the length, aifLi^ is held constant. Bar 1 is
excited by a distributed force Fi{xi,t).
3.1 Nominal transmitted power
The power transmitted from Bar 1 to Bar 2 for the nominal system (no
disorder) is briefly presented here. A more detailed derivation is shown in
the Appendix (see also Refs. [7] and [8]). The equations of motion are
Wi(a:i,t) -f ^Ci(wi(a;i,f)) = Fi(a:i,t)-|-
^[W2(a2, t)-wi(ai, - ci)
(jBo2A2-§^ + m2|^^W2(x2,t) + ^C'2(w2(a;2,t)) =
k[wi{ai,t)-W2{a2,t)]6{x2 - ^2)
1182
a)
Bar 2
Fig. 1: Two-bar system
where ^ is a Dirac delta function, and (for Bar i) Eoi is Young’s modulus,
Ai is the cross-sectional area, rm is the mass per unit length, ’Ci is the
viscous damping operator, and Wt(a:i,t) is the deflection. The deflections
of the two bars can be expressed by a summation of modes:
Wi{xi,t) = W2(x2,t) = E 1^2r(i)^2r(^2) (5)
t=0 r=0
where and W2^{t) are modal amplitudes, and and ^2r(^2)
are mode shape functions of the decoupled bars. These mode shape func¬
tions are normalized so that each modal mass is equal to the total mass of
the bar, Mi. Applying modal analysis and taking a Fourier transform, the
following equations are obtained:
Mi(f)ijWij = 4- fc^ij(ai)[E kF2r^2r(<^2) - E
_ (6)
M2<f>2,W2, = ^^^2,(<22)[E VFi,^i,(ai) - E W2.^2M2)]
i=0 r=0
[ul. - -f • 2Cia;i .a;)(2 - sgn(j))
y/^ • 2C2<^2.^)(2 - sgn(s))
1 for ^ > 0
0 for 2 = 0
where an over-bar (") denotes a Fourier transform, and are resonant
frequencies, (i and (2 are damping ratios, and is a modal driving force.
Mode 0 is a rigid body mode, which is why the sgn(i) term is present.
Note that the damping ratio of each bar is assumed to be the same for all
modes.
Next, the modal driving forces are assumed to be incoherent, and each
spectral density function is assumed to be constant (white noise) over a
finite range of frequency [2]. After some algebra, the power transmitted
<^2^ =
sgn(0 =
1183
(7)
from Bar 1 to Bar 2, 1112(0;), is found as:
2o;^PC2«^PlPl ^ ^2r(^2)0^2r
M1W2IAI2 ^ |(^2.|2
-^1 -^2 <?^2r
where 5pjpi is the same uniform spectral density function for each modal
driving force on Bar 1.
1112(0;) =
A =
3.2 Monte Carlo Energy Method (MCEM)
The disordered case is now considered, where each bar has a random length.
The ensemble-averaged transmitted power for a population of disordered
two-bar systems is found by tahing the expected value of Eq. (7):
^[ni2]
(8)
where Ni is the number of modes taken for Bar i (this is an arbitrary set of
modes that have been aliased to the numbers 1,2,- • ’,Ni). Since a truncated
set of modes is used, Eq. (8) is an approximation. The random variables
in Eq. (8) are the resonant frequencies of the bars (which are present in
the terms ^i;, ^2r? s,nd A).
Equation (8) may be solved numerically using a Monte Carlo method:
the random variables are assigned with a pseudo-random number generator
for each realization of a disordered system, and the transmitted power
is averaged for many realizations. This is called a Monte Carlo Energy
Method (MCEM) here. It may be used as a benchmark for comparing the
accuracy of other approximate methods.
Note that the resonant frequencies of a bar may be found directly from
the disordered length. Therefore, for the MCEM results in this study, the
actual number of random variables in Eq. (8) is taken to be one for each
decoupled bar. That is, the two random lengths are assigned for each real¬
ization, and then the natural frequencies are found for each bar in order to
calculate the transmitted power. If such a relation were not known, each
resonant frequency could be treated as an independent random variable.
3.3 SEA-equivalent Transmitted Power
An SEA approximation of the transmitted power may be obtained by ap¬
plying several typical SEA assumptions to Eq. (8). (Since Eq. (1) is not
used directly, this might be called an SEA-equivalent transmitted power.)
These assumptions were summarized in Ref. [8]: the coupling between
1184
substructures is weak, the modal responses are uncorrelated, the expected
value of the square of mode shape functions at connection positions is unity,
and the pdfs of the resonant frequencies are uniform and identical.
The assumption of weak coupling means that the value of jAj in Eq.
(8) is approximately one. Applying the second and third assumptions then
yields
E[ai2]
M?M2
(9)
Since the pdf of each resonant frequency is assumed to be uniform, the
expected values in Eq. (9) are
E[
E[
l/4u)"
1
.2cos|
-f 2a;a;i. • cosf +
^ ul. — 2u}ujii • cos| •
• sin I
sin%
ian~
U)
2 _
(10)
l/4a;"C2Vr^, - 1)
- 1 - tan — — - - —
W2r=‘*'2ru
— ^2r
4w^C2\/i - Cl
(11)
W2r=‘*'2rj
where a = cos~^(l - 2Cj), subject to the restrictions (1 - 2(1)^ < 1 and
(1 _ 2(2)2 ^ I Finally, since the pdfs of the resonant frequencies are taken
to be identical, the frequency limits do not depend on the individual modes
(a;i. = uj2r, - a^nd = u;2,„ = oju)- Therefore, each sum in Eq. (9)
simplifies to the product of the expected value and the number of modes
in the frequency range of interest:
E[n^2]
2C2A:ViViiV25,^p^ r.r 1 1 Pr ^2. ^
WM2 Vi.r
(12)
Equation (12) is the SEA approximation used in this study.
3.4 Example
The three formulations of the transmitted power presented thus far the
nominal transmitted power in Eq. (7), the MCEM transmitted power in
Eq. (8), and the SEA-equivalent transmitted power in Eq. (12) — are
now compared for a two-bar system with the parameters shown in Table 1.
For the MCEM results, the disorder (ei and 62) was taken to be uniformly
distributed with mean zero and standard deviation ai = 0-2 = 10%.
As a measure of the coupling strength, the coupling ratio, Ri, is defined
cLS the ratio of coupling stiffness to the equivalent stiffness of a bar at the
1185
Table 1: Material properties and dimensions of two bars
MIIM2
21.53/21,53
[Kg]
R01IL02
10.58/8.817
[m]
E01IR02
200 XIOV2OO xlO^
[N/m'1
ai/a2
2.116/7.053
N
Pllp2
7,800/7,800
[Kg/m'*l
^pipi
1
[N^l
C1/C2
0.005/0.005
k
4.868 xlO^
[N/m;
Fig. 2: Comparison of the nominal transmitted power, the MCEM results
(20,000 realizations with (j^ = <72- 10%), and the SEA approximation.
fundamental resonant frequency, Ri = weak cou¬
pling is considered here such that Ri = 0.01.
The nominal transmitted power, the MCEM results, and the SEA ap¬
proximation are shown in Fig.2. The transmitted power calculated for
the nominal system exhibits distinct resonances. This is due to the low
modal overlap of the bars in this frequency range. The MCEM results
show distinct peaks for uj < 15,000 rad/s, but they become smooth as
the frequency increases. The SEA approximation does not capture indi¬
vidual resonances. However, at the higher frequencies where the disorder
effects are stronger, the SEA approximation agrees well with the MCEM
results. The frequency range between where the MCEM results are close
to the nominal results and where they are close to the SEA results (ap¬
proximately 2,500 ~ 15,000 rad/s for this case) is considered to be the
mid-frequency range here. This range will vary depending on the system
1186
parameters and the disorder strength. In the next section, an efficient ap¬
proximation of the transmitted power is presented which compares well
with MCEM in the mid-frequency range.
4 Parameter-based Statistical Energy
Method (PSEM)
The SEA approximation presented in the previous section does not capture
the resonances in the transmitted power because of two assumptions: the
resonant frequencies all have the same uniform pdf, and the values of the
square of mode shape functions at the connection positions are taken to be
the ensemble- averaged value. Keane proposed an alternate pdf of resonant
frequencies in order to apply SEA to the case of two coupled nearly periodic
structures [9]. This pdf is shown in Fig.3(a). It accounts for the fact that
Fig. 3: (a) The pdf of the natural frequencies and the resultant transmitted
power from Ref. [9]. (b) The pdfs of three natural frequencies, and a
schematic representation of the piecewise evaluation of transmitted power
for PSEM. The individual modal contributions are extrapolated ( - ) and
summed to calculate the total transmitted power ( — ).
the natural frequencies of a nearly periodic structure tend to be grouped in
several distinct frequency bands. Thus the pdf has a large constant value
for those frequency bands, and a small constant value elsewhere. The SEA
approximation of transmitted power is then modified by simply adding a
positive value or negative value on a logarithmic scale, as demonstrated in
Fig, 3(a). This solution thus captures some of the resonant behavior of the
transmitted power.
1187
Here, a more general approach is taken for approximating the power
transmitted between two substructures in a frequency range in which they
have low or intermediate modal overlap. Each resonant frequency is as¬
signed a uniform pdf. However, the frequency range of each pdf is diiferent;
it corresponds to the range in which that resonant frequency is most likely
to be found. (The concept of using “confidence bands” as one-dimensional
pdfs was suggested but not pursued in Ref. [8].) An example is shown in
Fig. 3(b) for three resonant frequencies. Furthermore, it is assumed that
the values of the square of the mode shape functions at the connection
positions are known. Thus, applying only the first two SEA assumptions
along with those noted above, Eq. (8) becomes:
N2
i:kme
This is called a Parameter-ba.sed Statistical Energy Method (PSEM) be¬
cause it employs information for individual modal parameters.
Since each modal pdf is uniform, Eqs. (10) and (11) still hold for the
expected values in Eq. (13). However, unlike the SEA approximation, each
expected value is different, because the corresponding frequency bounds are
unique. Furthermore, note that the pdfs do not cover the entire frequency
range of interest. The results for each mode are therefore extrapolated
outside the frequency range of that modal pdf before the individual modal
contributions are summed. This is shown schematically in Fig. 3(b). PSEM
is therefore a piecewise evaluation of the expected value of transmitted
power.
The PSEM approximation is now applied to the two-bar system of
Table 1, with the standard deviation of disorder cti = 0-2 = 10%. The pdfs
of the resonant frequencies of Bar 1 and Bar 2 aze shown in Fig. 4(a) and
(b), respectively. For this system, the bounds for each resonant frequency
may be found directly from the variation of the uncertain parameter. It
can be seen that the spread of each natural frequency pdf due to disorder
increases with increasing frequency.
The MCEM, PSEM, and SEA approximations for the transmitted power
are shown in Fig. 4(c). There were 20,000 realizations taken for the MCEM
results at each sampled frequency. This took about 10 hours of computa¬
tion time. In contrast, the PSEM results only required 3 seconds of compu¬
tation time, and the SEA results only required about 1 second. Note that
the PSEM results show excellent agreement with the much more expensive
MCEM results. The difference at very low frequencies comes from the fact
that for the PSEM approximation, the value of the term |A| was assumed
to be one due to weak coupling. This assumption breaks down as the
frequency approaches zero. However, the match between the MCEM and
1188
Fig. 4: (a) Natural frequency pdfs for Bar 1. (b) Natural frequency pdfs
for Bar 2. (c) Transmitted powers obtained by MCEM, PSEM, and SEA
for c7i = <J2 = 10%.
PSEM results in the mid-frequency range is excellent. Again, it is noted
that the SEA results converge to those of MCEM (and PSEM) as the fre¬
quency increases. Now it can be seen that the assumption of identical pdfs
for all modes becomes better with increasing frequency.
Next, the example system is considered with smaller disorder, Ci =
(72 = 1%. Fig. 5 shows the results for this case for what might be called
the mid-frequency range. Note that even though this is a higher frequency
range than that considered for the previous ca.se, the pdfs of the resonant
frequencies shown in Fig. 5(a) and (b) are not as strongly overlapped.
Thus, several peaks are seen in the transmitted power in Fig. 5(c). Again,
the PSEM approximation agrees well with the MCEM results, although
there is more discrepancy for this case. The SEA approximation follows
the global trend, but does not capture the resonances or anti-resonances.
The SEA results drop off at the edges because only modes within this
frequency range are considered to contribute to the transmitted power.
In addition to PSEM, another piecewise evaluation of the transmitted
power is considered here. For this approximation, wherever the individual
mode pdfs overlap, they are superposed to form a pdf for all the modes in
that “section” of the frequency range. This superposition is demonstrated
in Fig. 6. Also, if the number of modes in a section is above a certain cutoff
number, Nc, then it is assumed that their mode shape function values at
the connection positions are unknown, so that the ensemble-averaged value
must be used. This is called a multiple mode approximation. The purpose
1189
Fig. 5: (a) Natural frequency pdfs for Bar 1. (b) Natural frequency pdfs
for Bar 2. (c) Transmitted power obtained by MCEM, PSEM,-and SEA
for cTi = 0-2 = 1%.
Fig. 6: Resonant frequency pdfs for PSEM and for the multiple mode
approximation.
of formulating this approach is to investigate what happens as information
about the individual modes is lost.
The multiple mode approximation is applied to the example system
with cTi = <72 = 10% in Fig. 7. For A/c = 2, this approximation has good
agreement with MCEM. The match is especially good for u) < 10,000.
Above this frequency, the number of overlapped resonant frequencies in
each pdf section is greater than Nc, and the loss of mode shape information
affects the results slightly. For Nc = 0, the values of the mode shape
functions are taken to be one for the entire frequency range, just as in the
SEA approximation. As can be seen in Fig. 7, the piecewise construction
of the pdf roughly captures the frequency ranges of the resonances and
anti-resonances. However, the mode shape effect is more pronounced in
1190
-B.S
- MCEM
-e.s — Multiple mode approximation rvt= 2
. Multiple mode approximation Nfc= O
Fig. 7: Transmitted power obtained from MCEM and the multiple mode
approximation for Ui = <72 = 10%
the mid-frequency range. The peak values are now similar to the SEA
approximation.
5 Conclusions
In this study, the power transmitted between two multi-mode substruc¬
tures coupled by a spring was considered. A Monte Carlo Energy Method
(MCEM) was used to calculate the ensemble average of the transmitted
power for the system with parameter uncertainties. A classical Statistical
Energy Analysis (SEA) approximation matched the Monte Carlo results
in the high-frequency range, but did not capture the resonant behavior of
the transmitted power in the mid-frequency range where the substructures
have weak modal overlap.
A Parameter-based Statistical Energy Method (PSEM) was presented
which uses a distinct pdf for each natural frequency as well as some indi¬
vidual mode shape information. A piecewise evaluation of the transmitted
power was performed, and then the modal contributions were extrapolated
and superposed. The PSEM approximation compared very well with the
much more expensive Monte Carlo results, including in the mid-frequency
1191
range.
References
1 R. H. Lyon. Statistical Energy Analysis of Dynamical Systems: Theory
and Applications. M.I.T. Press, 1st edition, 1975.
2 R. H. Lyon. Theory and Application of Statistical Energy Analysis.
Butterworth-Heinemann, 2nd edition, 1995.
3 C. H. Hodges and J. Woodhouse. Theories of noise and vibration trans¬
mission in complex structures. Rep. Prog. Physics, 49:107-170, 1986.
4 J. Woodhouse. An approach to the theoretical background- of statis¬
tical energy analysis applied to structural vibration. Journal of the
Acoustical Society of America, 69(6):1695-1709, 1981.
5 M. P. Norton. Fundamentals of Noise and Vibration Analysis for En¬
gineers. Cambridge University Press, 1st edition, 1989.
6 N. Lalor. Statistical energy analysis and its use as an nvh analysis tool.
Sound and Vibration, 30(l):16-20, 1996.
7 Huw G. Davies. Power flow between two coupled beams. Journal of
the Acoustical Society of America, 51(1):393-401, 1972.
8 A. J. Keane and W. G. Price. Statistical energy analysis of strongly
coupled systems. Journal of Sound and Vibration, 117(2):363-386,
1987.
9 A. J. Keane. Statistical Energy Analysis of Engineering Structures
(Ph.D Dissertation). Brunei University (England), 1988.
Appendix
In this appendix, the nominal transmitted power in Eq. (7) for the mono-
coupled two-bar system is derived. The procedure follows that of Refs [7,
8].
Plugging Eq. (5) into Eq. (4),
F^ixut) + fcf E - Oi) (A.1)
t=0
1192
k £ Wi,(<)$i,(a,) - £ lV2,(«)«2,(a2) 5(x2 - aj)
Multiplying Eq. (A.l) by and integrating with respect to xi for
[0, Li] yields
MiK + =
A, + f f; W2,(4)W2Xo2) - E
Li
2CijUijMi = y C'i(«'i,.(a:i))Wij(xi)cia:i , fi, J Fi{xi,t)^i.{xi)dx],
0 0
and wij is the ;th resonant frequency of decoupled Bar 1. The damping
ratio in Eq. (A. 3) is now assumed to be the same (Ci) for all modes, since
the differences in the ratio are usually small and this simplifies the equation.
Taking the Fourier transform of Eq. (A.3) with zero initial conditions leads
to the following
= 7i, + S:$i,(ai)fETr2, $2,(02)
t r=0 i=0 -*
</>!;• = K, - • 2(iu;i^iv){2 - sgn(i))
J fori = 0
where (~) denotes a Fourier transform. Similarly, applying the previous
procedure to Eq. (A.2),
r OO CO -1
M2(^2.1V2. = ^^$2.(02) EW"li®l.(<»0-EW"2,$2,(o2) (A.5)
t i=0 r=0
^2s = {^2, - • 2C2^^2,w)(2 - sgn(s)).
Solving for VF2, from Eq. (A.4) and (A.5),
= Trr-Wx-l^^- - (a.6)
where
_ W2.(a2).^/ii$i,(ai)
1193
Calculating the second term in brackets in Eq. (A. 6),
fc4'2.(<22)EW^2,«'2.(a2) E
where
Plugging Eq. (A. 7) into Eq. (A. 6),
M2<j>2s 1 + CKl + Q;2 1=0
\ I ' I. -I— ✓
Coupling force Ps
Using the definition of transmitted power in Ref. [7], 1112(0)) is
ni2 = Re[-v^o)f;E[P.F;,(o))]
L „_ri
"I" 0^1 + P i~0 j=:0 J
A
i<?^2.p hh Ml,
where Re[] denotes the real part of the argument, * is a complex conjugate,
and
Finally, it is assumed that the modal driving forces, f-^., are uncorrelated.
Also, the spectral density function of each modal driving force is assumed
to be constant for the finite frequency range of interest:
C ( \ f ^PlVl ^
V for i^j.
Therefore, Eq. (A. 9) becomes
2^2fc^O)^.5^pipi ^ ^ ^2r(^2)o)2r
to to
(A.IO)
(A.ll)
1194
PASSIVE AND ACTIVE CONTROL III
Research on Control Law of Active Siispension of a
Seven Degree of Freedom Vehicle Model
Dr&Prof. YuchengLei Lifen Chen
Automobile Engineering Dept, Tong Ji University ,Shang
Hai,P.R.of China
Abstract
In the paper , control law of active suspension is presented ,
which involves 7-DOF vehicle model for improving control
accuracy .The control law involve vehicle running velocity ,
road power spectrum , suspension stiffness and damping .The
control law can be applied to multi-DOF control of active
suspension of vehicle .
Keywords: Active suspension , control law , Game theory ,
Modeling , 7 — DOF Vehicle Model .
1 . Introduction
An individual control system for each wheel by applying the
optimum regulation method for the two degrees of freedom is
showed in [3] . [4] and [5] also introduce two-DOF feedback
control method of active suspension .It is difficult for two-
DOF control method to coordinate multi-DOF kinematic
distances of entire car . Muti-DOF active control can improve
coordination control accuracy of entire car , but high speed of
CPU is asked for control and calculation while control law of
multi-DOF is got by real-time calculation . And, ride
performance and handling performance is inconsistent . For
resolving the problem ,the paper holds a new calculation
method for optimizing the law that can be programmed for
real-time control by table-lookup and not by real-time
calculation .So the method and law can not only improve
coordination control accuracy ,but also develop control
speed .
2 . Mathematical Model
Vehicle is simplified to turn into 7 DOF model . 7-DOF
vibration motion equation can be written as follow
[Af ]z+ [c]z+ [kY = \C, ](2+ ]e (1)
1195
Where [M] is mass matrix , [C] is suspension damping
matrix , [A’] is suspension stiffness matrix , [C,] is tyre
• •
damping matrix , \K, ] is tyre stiffness matrix , Z is
acceleration matrix ,Z is velocity matrix , Z is 7-DOF
displacement matrix , Q is road surface input velocity matrix ,
Q is road surface input displacement matrix .
z=p„z....,zy (2)
•• . • -j-J
Where Zi is vehicle vertical acceleration, Zi is roll
•• ••
acceleration , Z3 is pitch acceleration , Z,- (/ = 4,*«*,7) is
four tyres vertical acceleration .
3 General Optimization Method of Control Law
Objective function of optimization of control law can
generally got by calculating weighted sum of 7-DOF mean
square root of acceleration , dynamic deflection and dynamic
load .it can be written as follows
/=! >1 A=1
Where a-^ (z = !,• • (7 = !,• • •A), r * = h* • •A) is
weighted ratio . Where a.. (i=l, * * ' ,7) is 7-DOF mean
Zi
square root of acceleration , o-jy_ 0=1? ‘ ' is 7-DOF mean
square root of dynamic deflection , (k=l, * * ^ A) is
7-DOF mean square root of relative dynamic load .cr.. ,
Zi
can be calculated by resolving (1) using numerical
method .
4 Result of General Optimization
Method of Control Law
Optimization result of control law of a truck is got using
above method as figure 1 and 2 , its main parameters as
follows .
Wheel distance is 1.4 meter , axle distance is 2.297 meter ,
mass is 1121.3 kg , front tyre and axle’ s mass is 22.8 kg ,
rear tyre and axle’ s mass is 35.0 kg , X axis’ rotational
1196
inertia is 307.4 kg-ni^ , Y axis’ rotational inertia is 1276.5
kg-rri^ .
In fig. 1 and 2 , Cl of RMSMIN and C3 of RMSMIN are
respectively front and rear suspension damping of getting
minimization of above objective function , it is changing
while road surface rough coefficient and automobile
velocity V is changing . Cl and C3 also rise when velocity V
rises . This is called control law of general optimization
method of active suspension in the paper . The result in fig. 1
and fig. 2 has been verified by road test .
Fig. 1 front suspension Fig. 2 rear suspension
optimization damping optimization damping
Simulation result can also verify that ride performance’s
increasing (suspension stiffness reducing) will make handling
performace reduce . So selecting perfect
= is very difficult
and inconsistent .The paper advances next game method to
try to resolve the inconsistent problem .
5 Game Optimization Method of Control Law
Because to select weighted ratio of general optimization
method is difficult , the paper advances a new method of
optimization of control law — Game Balance Optimization
Method 4t is discussed as follows .
Game theory method of two countermeasure aspect can be
expressed as follows :
1197
— 1 j ^ 0 ,z •— 1, 2,
tr (4)
^ hj — I , hj ^0 , y = 1, 2,- • *, WZ2
where . r^, is probability of selecting R^,R2> . of
countermeasure R ( where R is acceleration mean square
root ) , and h,,h^, . X, is probability of selecting
. , of countermeasure H ( where H is
mean square root of deflection or handling and
satiability ) .
It is called hybrid game method while these probability is
leaded into the method . Countermeasure R selects in order
to get maximization of minimization paying expected value
of column vector of paying matrix , and countermeasure H
selects hj in order to get minimization of maximization
paying expected value of row vector of paying matrix .
If rank of paying matrix is x ,R should select r., as
follows.
max\min\
(5)
relative
And H should select hj as follows :
MAXl MINI
j=i
(6)
relative hj
a.j (z = l, . ,y = l, . ,^2) in (5) and (6) is element
value of paying matrix , basing vehicle theory it can be got
as follows :
a.j = C, ! + (7)
Where C^,C^ in (7) is coefficient of paying matrix ( The
paper orders they is 1 as an example , as Q ,C2 ’ s real value
about very much condition is related to some privacy
1198
problem it can’ t be introduced. ) . (5), (6) called respectively
minimization maximization expected value and maximization
minimization expected value can be abbreviated as
MAxi^IN^^ )} and MIN ^AX(^ )} . if r.,andhj is
got as optimization of countermeasure , it can be wntten as
follows :
MAX^IN^ ^^Optimization Countermeasure
Expected Value ^ MIN
A probability association {r^ Xj) can be content with
optimization expected value as follows :
Optimization Countermeasure Expected
m^ m-i
Value=|] J^aij»r;»hj (9)
/=!
Writer advances reformation simplex algorithm for resolving
the game problem as reference [1] . In the paper the water
selects only an example to introduce calculation results as
follows because the paper has limited space .
6 Results of Game Optimization Method of Control Law
Paying matrix as fig. 3 and fig. 4 , optimization result of
control law of a truck is got using above game method as
figure 5 and 6 , calculated truck’s main parameters as
follows .
Wheel distance is 1.23 meter , axle distance is 3.6 meter ,
mass is 13880.0 kg , front tyre and axle’ s mass is 280.0 kg ,
rear tyre and axle’ s mass is 280.0 kg , X axis’ rotational
inertia is 1935 kg^m^ , Y axis’ rotational inertia is 710
kg-m- .
In fig. 3 and 4 3(IJ) is paying matrix value
{i = 1, . ,m, J = 1, . ,m,) . In fig. 5 and fig. 6 , K1 of
RMSMIN and K3 of RMSMIN are respectively front and
rear suspension stiffness of getting optimization
countermeasure expected value of above game method , it is
changing while road surface rough coefficient Q and
automobile velocity V is changing . K1 and K3 also rise
1199
when velocity V rises . This is called control law of active
suspension . The result in fig. 5 and fig. 6 has been verified
by road test .
Fig. 3 paying matrix Fig. 4 paying matrix
7 Conclusion
The paper introduces two method to get optimization control
Fig. 5 front suspension Fig. 6 rear suspension
optimization stiffness optimization stiffness
law of active suspension , and the control law is verified to
ability to be applied to real control of active suspension . This
will develop control accuracy and speed of active
suspension .Off course , it need being researched further .
8 reference
1 Lei Yucheng , Theory and Engineering Realization of
Semi — Active Control of Vehicle Vibration ,Dr. paper ,
Harbin Institute of Technology , China ,1995 6 .
2 Thompson A.G. , A Suspension Proc. Int of Mechanical
1200
Engr. Vol 185 No. 36, 970 — 990,553 563 .
3 Lei , S. , Fasuda , E. and Hayashi , Y. : “An Experimental
Study of Optimal Vibration Adjustment Using Adaptive
Control Methods ”, Proc . IMechE Int . Conf . Advanced
Suspensions , London , England , (1988) , C433/88 , 119-
124.
4 Kamopp D. , Active Damping in Road Vehicle Systems ,
VSD, 12(1983), 291-316.
5 Kamopp D . C ., Grosby M. J. & Harword R. ,Vibration
Control Using Semi-Active Force , Generator , Trans .
ASME, J . Eng . for Ind . Vol . 96 Ser . B , No .2 , (1974) ,
619-626 .
1201
1202
Designing Heavy Truck Suspensions for Reduced Road Damage
Mehdi Ahmadian
Edward C. Mosch Jr.
Department of Mechanical Engineering
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061-0238; USA
(540) 231-4920/-9100(fax)
ahmadian@vt.edu
ABSTRACT
The role of semiactive dampers in reducing tire dynamic loading is examined.
An alternative to the well-known skyhook control policy, called
“groundhook,” is introduced. Using the dynamic model of a single suspension,
it is shown that groundhook semiactive dampers can reduce tire dynamic
loading, and potentially lessen road damage, for heavy trucks.
INTRODUCTION
The main intent of this work is to determine, analytically, the role of
semiactive suspension systems in reducing tire dynamic loading, and road and
bridge damage. Although primary suspension systems with semiactive
dampers have been implemented in some vehicles for improving ride and
handling, their impact on other aspects of the vehicle remain relatively
unknown. Specifically, it is not yet known if implementing semiactive
dampers in heavy truck suspension systems can reduce the tire dynamic
forces that are transferred by the vehicle to the road. Reducing dynamic forces
will result in reducing pavement loading, and possibly road and bridge damage.
The idea of semiactive dampers has been in existence for more than
two decades. Introduced by Karnopp and Crosby in the early 70’ s [1-2],
semiactive dampers have most often been studied and used for vehicle primary
suspension systems. A semiactive damper draws small amounts of energy to
operate a valve to adjust the damping level and reduce the amount of energy
that is transmitted from the source of vibration energy (e.g., the axle) to the
suspended body (e.g., the vehicle structure). Therefore, the force generated by
a semiactive damper is directly proportional to the relative velocity across the
damper (just like a passive damper). Another class of dampers that is usually
considered for vibration control is fully active dampers. Active dampers draw
1203
relatively substantial amounts of energy to produce forces that are not
necessarily in direct relationship to the relative velocity across the damper.
The virtues of active and semiactive dampers versus passive dampers
have been addressed in many studies [3-10]. Using various analytical and
experimental methods, these studies have concluded that in nearly all cases
semiactive dampers reduce vibration transmission across the damper and
better control the suspended (or sprung) body, in comparison to passive
dampers. Further, they have shown that, for vehicle primary suspension
systems, semiactive dampers can lower the vibration transmission nearly as
much as fully active dampers; without the inherent cost and complications
associated with active dampers. This has led to the prototype application,
and production, of semiactive dampers for primary suspensions of a wide
variety of vehicles, ranging from motorcycles, to passenger cars, to bus and
trucks, and to military tanks, in favor of fully active systems.
Although there is abundant research on the utility of semiactive
dampers for improving vehicle ride and handling, their potential for reducing
dynamic forces transmitted to the pavement remains relatively unexplored.
This is because most suspension designers and researchers are mainly
concerned with the role of suspension systems from the vehicle design
perspective. Another perspective, however, is the effect of suspension
systems on transmitting dynamic loads to the pavement.
ROAD DAMAGE STUDIES
Dynamic tire forces, that are heavily influenced by the suspension, are
believed to be an important cause of road damage. Cole and Cebon [11]
studied the design of a passive suspension that causes minimum road damage
by reducing the tire force. They propose that there is a stronger correlation
between the forth power of the tire force and road damage than the dynamic
load coefficient (DLC) and road damage.
A simple measure of road damage, introduced by Cebon m [12], is the
aggregate fourth power force defined as
Na
Al=Y,Pjk k= 1,2,3 ..ns (1)
where Pjk = force applied by tire j to point k along the wheel path,
ria - number of axles on vehicle, and
Us = number of points along the road..
1204
DLC is a popular measurement frequently used to characterize dynamic
loading and is defined as the root mean square (RMS) of the tire force divided
by the mean tire force, which is typically the static weight of the vehicle. The
equation takes the form;
RMS Dynamic Tire Force
Static Tire Force
This study shows that minimum road damage, for a two degree-of-freedom
model, is achieved by a passive system with a stiffness of about one fifth of
current air suspensions and a damping of about twice that typically provided.
In practice, however, reducing the suspension stiffiiess can severely limit the
static load carrying capacity of the suspension and cause difficulties in vehicle
operation. Further, higher damping can substantially increase vibration
transmission to the body and worsen the ride.
In another study by Cole and Cebon [13] a two-dimensional articulated
vehicle simulation is validated with measurements from a test vehicle. The
effect of modifications to a trailer suspension on dynamic tire forces are
investigated. The RMS of dynamic loads generated by the trailer are predicted
to decrease by 3 1 per cent, resulting in a predicted decrease in theoretical road
damage of about 13 per cent.
Yi and Hedrick compared the effect of continuous semiactive and
active suspensions and their effect on road damage using the vehicle simulation
software VESYM [14]. A control strategy based on the tire forces in a heavy
truck model is used to show that active and semiactive control can potentially
reduce pavement loading. They, however, mention that measuring the tire
forces poses serious limitation in practice.
The primary purpose of this paper is to extend past studies on
semiactive suspension systems for reducing road damage. An alternative
semiactive control policy, called "groundhook,” is developed such that it can
be easily applied in practice, using existing hardware for semiactive
suspensions. A simulation model representing a single primary suspension is
used to illustrate the system effectiveness. The simulation results show that
groundhook control can reduce the dynamic load coefficient and fourth power
of tire force substantially, without any substantial increase in body
acceleration.
1205
MATHEMATICAL FORMULATION
We consider a model representing the dynamics of a single primary suspension
in a heavy truck, as shown in Figure 1.
Truck Mass
Suspension
Stiffness
Suspension
and Tire Mass
Tire
Stiffness
Suspension
Damping
Road Input
Figure 1. Primary Suspension Model
This model has been widely used in the past for automobile applications, due
to its effectiveness in analyzing various issues relating to suspensions.
Although it does not include the interaction between the axles and the truck
frame dynamics, the model still can serve as an effective first step in studying
fundamental issues relating to truck suspensions. Follow up modeling and
testing, using a full vehicle, is needed to make a more accurate assessment.
The dynamic equations for the model in Figure 1 are:
M^x^+C(x^-X2) + K(x^~X2) = 0 (3a)
MjX, - C(ii -X2)-K{x^ - ^2) + 0
The variables Xi and X2 represent the body and axle vertical displacement,
respectively. The variable Xjn indicates road input, that is assumed to be a
random input with a low-pass (0 - 25 Hz) filter. The amplitude for Xjn is
adjusted such that it creates vehicle and suspension dynamics that resembles
field measurements. Such a function has proven to sufficiently represent
actual road input to the vehicle tires.
Table 1 includes the model parameters, that are selected to represent a
typical laden truck used in the U.S. The suspension is assumed to have a
linear stiffness in its operating range. The damper characteristics are modeled
as a non-linear function, as shown in Figure 2.
1206
Table 1. Model Parameters
Description
Symbol
Value
Body Mass
Ml
287 kg
Axle Mass
M2
34 kg
Suspension Stiffness
K1
196,142 N/m
Tire Stifftiess
Kt
1,304,694 N/m
Suspension Damping
C
See Table 2
The bilinear function in Figure 2 represents the force-velocity
characteristics of an actual truck damper. The parameters selected for both
passive and semiactive dampers are shown in Table 2. These parameters are
selected based on truck dampers commonly used in the U.S. Although we
examined the effect of damper tuning on dynamic loading, if falls outside the
scope of this paper. Instead, we concentrate here on comparing different
semiactive dampers with a passive damper, using the baseline parameters
shown in Table 2. The semiactive and passive damper characteristics used for
this study are further shown in Figure 3.
1207
Force Velocity Curve
Figure 3. Passive and Semiactive Damper Characteristics
SKYHOOK CONTROL POLICY
As mentioned earlier, the development of semiactive dampers dates back to
early 70’s when Kamopp and Crosby introduced the skyhook control policy.
For the system shown in Figure 1, skyhook control implies
X,(X] - ;C2 ) - 0 C = Con
X](X^-X2)<0 C = Coff
Where x, and represents the velocities of Mi (vehicle body) and M2 (axle),
respectively. The parameters Con and Coff represent the on- and off-state of
the damper, respectively, as it is assumed that the damper has two damping
levels. In practice, this is achieved by equipping the hydraulic damper with an
orifice that can be driven by a solenoid. Closing the orifice increases damping
level and achieves Con, whereas opening it gives Coff .
1208
Table 2. Damper Parameters
Passive
Semiactive On-State
Semiactive Off-State
n
0.25
0.35
0.03
m
0.10
0.15
0.03
Vbb
0.254 m/sec
0.254 m/sec
0.254 m/sec
B
0.20
0.30
0.03
S4
0.10
0.15
0.03
Vbj
0.254 m/sec
0.254 m/sec
0.254 m/sec
The switching between the two damper states, shown in Eq. (4), is
arranged such that when the damper is opposing the motion of the sprung
mass (vehicle body), it is on the on-state. This will dampen the vehicle body
motion. When the damper is pushing into the body, it is switched to the off-
state to lower the amount of force it adds to the body. Therefore, a semiactive
damper combines the performance of a stiff damper at the resonance
frequency, and a soft damper at the higher frequencies, as shovm in Figure 4.
1209
Figure 5. Groundhook Damper Configurations: a) optimal groundhook
damper configuration, b) semiactive groundhook damper
configuration.
This feature allows for a better control of the vehicle body, as has been
discussed in numerous past studies. The skyhook control policy in Eq. (4),
however, works such that it increases axle displacement, X2, (commonly called
wheel hop). Because the tire dynamic loading can be defined as
DL = KtX2 (5)
The skyhook control actually increases dynamic loading. As mentioned earlier
the development of skyhook policy was for improving ride comfort of the
vehicle, without losing vehicle handling. Therefore, the dynamic loading of the
tires was not a factor in the control development.
GROUNDHOOK CONTROL POLICY
To apply semiactive dampers to reducing tire dynamic loading, we propose an
alternative control policy that can be implemented in practice using the same
hardware needed for the skyhook policy. To control the wheel hop, this
policy, called “groundhook,” implies:
X, (;ci - X2 ) < 0 C = Con
(6a)
x, (Xt - ^2 ) > 0 C = Coff
(6b)
As shovm in Figure 5, the above attempts to optimize the damping force on
the axle, similar to placing a damper between the axle and a fictitious ground
(thus, the name “groundhook”). The groundhook semiactive damper
maximizes the damping level (i.e., C = Con) when the damper force is opposing
1210
the motion of the axle; otherwise, it minimizes the damping level (i.e., C =
Coff). The damper hardware needed to implement groimdhook semiactive is
exactly the same as
the skyhook semiactive, except for the control policy programmed into the
controller.
SIMULATION RESULTS
The model shown in Figure 1 is used to evaluate the benefits of groundhook
dampers versus passive and skyhook dampers. A non-linear damper model
was considered for the simulations, as discussed earlier. The road input was
adjusted such that the dynamic parameters for the passive damper resembles
actual field measurements. Five different measures were selected for
comparing the dampers:
• Dynamic Load Coefficient (DLC)
• Fourth Power of the tire dynamic load
• Sprung mass acceleration
• Rattle Space (relative displacement across the suspension)
• Axle Displacement, relative to the road
Dynamic load coefficient and fourth power of tire force are measures
of pavement dynamic loading and are commonly used for assessing road
damage. They are both considered here because there is no clear consensus on
which one is a better estimate of road damage. Axle displacement, relative to
the road, indicates wheel hop and is directly related to DLC and tire force,
therefore it is yet another measure of road damage. Sprung mass acceleration
is a measure of ride comfort. Our experience, however, has shown that for
trucks this may not be a reliable measure of the vibrations the driver feels in
the truck. The relative displacement across the dampers relates to the rattle
space, that is an important design parameter in suspension systems,
particularly for cars. For each of the above measures, the data was evaluated
in both time (Figures 6-7) and frequency domain (Figures 8 - 10). In time
domain, the root mean square (RMS) and maximum of the data for a five-
second simulation are compared. In frequency domain, the transfer function
between each of the measures and road displacement is plotted vs. frequency.
The frequency plots highlight the effect of each damper on the body and axle
resonance frequency.
Figures 6 and 7 show bar charts of root mean square (RMS) and
maximum time data, respectively. In each case the data is normalized with
respect to the performance of passive dampers commonly used in trucks.
Therefore, values below line 1 .0 can be interpreted as an improvement over the
existing dampers. As Figures 6 and 7 show, groundhook dampers significantly
1211
improve pavement loading, particularly as related to the fourth power of tire
force. Furthermore, the rattle space is improved slightly over passive
dampers, indicating that groundhook dampers do not impose any additional
burden on the suspension designers.
One measure that has increased due to groundhook dampers is body
acceleration. As mentioned earlier, in automobiles this measure is used as an
indicator of ride comfort. In our past testing, however, we have found that for
trucks it is a far less accurate measure of ride comfort. This is mainly due to
the complex dynamics of the truck frame and the truck secondary suspension.
A more accurate measure of ride comfort is acceleration at the B-Post (the
post
Tire Axle Body Rattle Space
Dynamic Displacment Acceleration
Force ^ 4
Figure 6. RMS Time Data Normalized with respect to Passive Damper
Tire Axle Body Rattle Space
Dynamic Displacment Acceleration
Force 4
Figure 7. Max. Time Data Normalized with respect to Passive Damper
1212
behind the driver), which cannot be evaluated from the single suspension
model considered here. Nonetheless, the body acceleration is included for the
sake of completeness of data.
The model shows that skyhook dampers actually increase the
measures associated with pavement loading, while improving body
acceleration. This agrees with the purpose of skyhook dampers that are
designed solely for improving the compromise between ride comfort and
vehicle handling. The improvement in ride comfort occurs at the expense of
increased pavement loading.
Figure 8. Transfer Function between Axle Displacement and Road Input
Figures 8-10 show the frequency response of the system due to each damper.
In each figure, the transfer fimction between one of the measures and input
displacement is plotted vs. frequency. These plots highlight the impact of
skyhook and groundhook on the body and wheel hop resonance, relative to
existing passive dampers. The frequency plots indicate that the
1213
Transfer
Frequency (Hz)
groundhook dampers reduce axle displacement and fourth power of tire
dynamic force at wheel hop frequency. At body resonance frequency,
groundhook dampers do not offer any benefits over passive dampers. The
frequency results for body acceleration and rattle space are similar to those
discussed earlier for the time domain results. The frequency plots show that
the skyhook dampers offer benefits over passive dampers at frequencies close
to the body resonance frequencies. At the higher frequencies, associated with
wheel hop, skyhook dampers result in a larger peak than either passive or
groundhook dampers. This indicates that skyhook dampers are not suitable
for reducing tire dynamic loading.
CONCLUSIONS
An alternative to skyhook control policy for semiactive dampers was
developed. This policy, called “groundhook,” significantly improves both
dynamic load coefficient (DLC), and fourth power of tire dynamic load,
therefore holding a great promise for reducing road damage to heavy trucks.
The dynamic model used for assessing the benefits of groundhook dampers
represented a single suspension system. Although the results presented here
show groundhook dampers can be effective in reducing tire dynamic loading
and pavement damage, more complete models and road testing are necessary
for more accurately assessing the benefits.
REFERENCES
1. Crosby, M. J., and Karnopp, D. C., "The Active Damper," The Shock
and Vibration Bulletin 43, Naval Research Laboratory, Washington, D.C.,
1973.
2. Karnopp, D. C., and Crosby, M. J., "System for Controlling the
Transmission of Energy Between Spaced Members," U.S. Patent
3,807,678, April 1974.
3. Ahmadian, M. and Marjoram, R. H., “Effects of Passive and Semi-active
Suspensions on Body and Wheelhop Control,” Journal of Commercial
Vehicles, Vol. 98, 1989, pp. 596-604.
4. Ahmadian, M. and Marjoram, R. H., “On the Development of a
Simulation Model for Tractor Semitrailer Systems with Semiactive
Suspensions,” Proceedings of the Special Joint Symposium on Advanced
Technologies, 1989 ASME Winter Annual Meeting, San Francisco,
California, December 1989 (DSC-Vol. 13).
5. Hedrick, J. K., "Some Optimal Control Techniques Applicable to
Suspension System Design," American Society of Mechanical Engineers,
Publication No. 73-ICT-55, 1973.
1215
6. Hac, A., "Suspension Optimization of a 2-DOF Vehicle Model Using
Stochastic Optimal Control Technique," Journal of Sound and Vibration,
1985.
7. Thompson, A. G., "Optimal and Suboptimal Linear Active Suspensions
for Road Vehicles," Vehicle System Dynamics, Vol. 13, 1984.
8. Kamopp, D., Crosby, M. J., and Harwood, R. A., "Vibration Control
Using Semiactive Force Generators," American Society of Mechanical
Engineers, Journal of Engineering for Industry, May 1974, pp. 619-626.
9. Krasnicki, E. J., "Comparison of Analytical and Experimental Results for
a Semiactive Vibration Isolator," Shock and Vibration Bulletin, Vol. 50,
September 1980.
10. Chalasani, R.M., "Ride Performance Potential of Active Suspension
Systems-Part 1: Simplified Analysis Based on a Quarter-Car Model,"
proceedings of 1986 ASME Winter Annual Meeting, Los Angeles, CA,
December 1986.
11. Cole, D. J. and Cebon, D., “Truck Suspension Design to Minimize Road
Damage,” Proceedings of the Institution of Mechanical Engineers, Vol.
210, D06894, 1996, pp. 95-107.
12. Cebon, D., “Assessment of the Dynamic Forces Generated by Heavy
Road Vehicles,” ARRB/FORS Symposium on Heavy Vehicle Suspension
Characteristics, Canberra, Australia, 1987.
13. Cole, D. J. and Cebon, D., “Modification of a Heavy Vehicle Suspension
to Reduce Road Damage,” Proceedings of the Institution of Mechanical
Engineers, Vol. 209, D03594, 1995.
14. Yi, K. and Hedrick, J. K., “Active and Semi- Active Heavy Truck
Suspensions to Reduce Pavement Damage,” SAE SP-802, paper 892486,
1989.
1216
Active Vibration Control of Isotropic Plates Using
Piezoelectric Actuators
A. M. Sadri", J. R., Wright* and A. S. Cherry*
The Manchester School of Engineering, Manchester M13 9PL, UK
and
R. J. Wynnes
Sheffield Hallam University, School of Engineering, Sheffield, UK
Abstract: Theoretical modelling of the vibration of plate components of a
space structure excited by piezoelectric actuators is presented. The equations
governing the dynamics of the plate, relating the strains in the piezoelectric
elements to the strain induced in the system, are derived for isotropic plates
using the Rayleigh-Ritz method. The developed model was used for a simply
supported plate. The results show that the model can predict natural
frequencies and mode shapes of the plate very accurately. The open loop
frequency response of the plate when excited by the patch of piezoelectric
material was also obtained. This model was used to predict the closed loop
frequency response of the plate for active vibration control studies with
suitable location of sensor-actuators.
Introduction
Vibration suppression of space structures is very important because
they are lightly damped due to the material used and the absence of air
damping. Thus the modes of the structure must be known very
accurately in order to be affected by the controller while avoiding
spillover. This problem increases the difficulty of predicting the
behaviour of the structure and consequently it might cause unexpected
on-orbit behaviour.
These difficulties have motivated researchers to use the
actuation strain concept. One of the mechanisms included in the
actuation strain concept is the piezoelectric effect whereby the strain
induced through a piezoelectric actuator is used to control the
Research Student, Dynamics &: Control Research Group.
^ Professor , Dynamics & Control Research Group,
^ Former Lecturer, Dynamics & Control Research Group.
^ Professor of Mechanical and Control Engineering.
1217
deformation of the structure [1]. It can be envisaged that using this
concept in conjimction with control algorithms can enhance the ability
to suppress modes of vibration of flexible structures.
Theoretical and experimental results of the control of a flexible
ribbed antenna using piezoelectric materials has been investigated in
[2]. An active vibration damper for a cantilever beam using a
piezoelectric polymer has been designed in [3]. In this study,
Lyapunov's second or direct method for distributed-parameter systems
was used to design control algorithms and the ability of the algorithms
was verified experimentally. These works have clearly shown the
ability of piezoelectric actuators for vibration suppression. However,
they have been limited to one dimensional systems. Obviously, there is
a need to understand the behaviour of piezoelectric actuators in two
dimensional systems such as plates.
Vibration excitation of a thin plate by patches of piezoelectric
material has been investigated in [4]. Their work was basically an
extension of the one dimensional theory derived in [1] to show the
potential of piezoelectric actuators in two dimensions. In their studies,
it was assumed that the piezoelectric actuator doesn't significantly
change the inertia, mass or effective stiffness of the plate. This
assumption is not guaranteed due to the size, weight and stiffness of
the actuator. Based on this assumption, their model can not predict the
natural frequencies of the plate accurately after bonding piezoelectric
actuators. Therefore, it is essential to have a more general model of a
plate and bonded piezoelectric actuators with various boundary
conditions. The model should be able to predict frequency responses
because this is fundamental to the understanding of the behaviour of
the system for control design purposes. It is the objective of the current
study to develop such a modelling capability.
Previous work [5, 6, 7] has concentrated on the modelling and
control of a cantilever beam. The method used involved bonding
piezoelectric material to a stiff constraining layer, which was bonded to
the beam by a thin viscoelastic layer in order to obtain both active and
passive damping. Then a Rayleigh-Ritz model was developed and
used to derive a linearized control model so as to study different
control strategies. In the work described in this paper, the method has
been extended to the more complex plate problem. The paper
introduces a modelling approach based on the Rayleigh-Ritz assumed
mode shape method to predict the behaviour of a thin plate excited by
a patch of piezoelectric material bonded to the surface of the plate. The
model includes the added inertia and stiffness of the actuator and has
been used to predict the frequency response of the plate. Suggestions
for future work are also included.
1218
Theoretical Modelling
In developing the Rayleigh-Ritz model of a plate excited by a
patch of piezoelectric material bonded to the surface of the plate, a
number of assumptions must be made. The patch of piezoelectric
material is assumed to be perfectly bonded to the surface of the plate.
The magnitude of the strains induced by the piezoelectric element is a
linear fimction of the applied voltage that can be expressed by
e:=E:
(1)
Here is the piezoelectric strain constant, is the piezoelectric layer
thickness and V33 is the applied voltage. The index 31 shows that the
induced strain in the '1' direction is perpendicular to the direction of
poling '3' and hence the applied field. The piezoelectric element
thickness is assumed to be small compared to the plate thickness. The
displacements of the plate middle surface are assumed to be normal to
it due to the bending affects.
Figure 1 shows the configuration of the bonded piezoelectric
material relative to the surface of the plate.
w
Figure 1. Configuration of the bonded piezoelectric actuator on the surface of the
plate.
In figure 1, 4 and 4 are the dimensions of the plate, x, , X2 , y, and y^
are the boundaries of the piezoelectric element and w, v and w are the
displacements in the x , y and z direction, respectively.
To derive the equations of motion of the plate based on the
Rayleigh-Ritz method, both the strain energy U and kinetic energy T of
the plate and the piezoelectric element must be determined. The strain
1219
and kinetic energy result from the deformation produced by the
applied strain which is induced by exciting the piezoelectric element.
The deformations can be expressed by the combination of the midplane
displacement and the deformations resulting from the bending of the
plate.
Strain Energy
The strain energy of the plate and piezoelectric material can be
calculated by
U = +T;«yY,<y)dVp +T^T„)dVp. (2)
where 8 is the inplane direct strain, a is the inplane direct stress, t is
the inplane shear stress and y is shear strain. dV shows volume
differential and indices p and pe refer to the plate and piezoelectric
actuator, respectively. The strains 8^ , 8^ and y can be shown to be
For the Plate :
du
dv d (3)
_du dv d^w
^ dy dx dxdy
For the Piezoelectric actuator :
Ev =
du
a
^~'dx~
^ a?*
Sv
a
du av
« ■ _ 1 ^
dy dx
'"33=S,-H'^33
a
dxdy ^
(4)
where _ refers to the strains due to the deformation. The stresses ,
Gy , y can be expressed as
1220
For the Plate :
E„
1
0
‘e. '
1
0
y.vy.
0
0
r.y_
2 .
(5)
For the Piezoelectric actuator :
'
0^.
II
1
0
Sx-H^33
1
0
r.y_
0
0
l-v
2 .
. Is
(6)
where E is Young modulus and v is Poisson's ratio for the assumed
isotropic material. Substituting equations 3, 4, 5 and 6 into 2 yields the
strain energy of the plate and piezoelectric actuator.
^ S, + sj + 1 ( 1 - Dp) dVp
- '"''33 )' + 2Dpp( e, - HV33 ){ E, - M.V33 )^ + ^ 1 ciVpp.
•'Vp, ^ _ u — - tC
(7)
Kinetic Energy
To obtain the kinetic energy, the velocity components in x, y and
z directions are needed. The velocity components can be calculated by
differentiating the displacement components which are
dw
u— u— z—
dx
dw
(8)
dy
w=w.
Differentiating equations 8 yields
1221
dw
u= u- z —
ax
dw
v= V- z —
dy
w
(9)
where u , y and w are the velocity components in the x, y and z
directions respectively. Using these velocity components, the kinetic
energies of the plate and piezoelectric actuator are obtained as
dw
P rw^ + (u-z^f + (v-z^^]dV^
9x
ay'
(10)
where p is the mass density.
Equation of Motion
The static or dynamic response of the plate excited by the
piezoelectric actuator can be calculated by substituting the strain and
kinetic energy into Lagrange's equation
d dT dT ^ dU
dt dq. dq,"^ dq.
(11)
where q^ represents the ith generalised coordinate and is the ith
generalised force. As there are no external forces (the force applied by
5\e piezoelectric element is included as an applied strain) or gyroscopic
terms and there is no added damping, Lagrange's equation reduces to :
dt dq, ^ dq,
(12)
Now the equation of motion can be obtained by using the expression
obtained for the strain and kinetic energy, and the assumed shape
functions for flexural and longitudinal motion
1222
u(x,y,t)~
v(x,y,t)={i,(x,y)y
w(x, y, t) = {(|)(j:, y)}^ {?(?)}.
(13)
Here y , ^ and ([) are the assumed displacement shape and h , f and g
are generalised coordinate of the plate response m x, y and z
directions. Using the shape functions expressed in equations 13,
substituting equations 7 and 10 into equation 12, and including
Rayleigh damping yields the equation of motion of the plate in the
form
where M, C and K are mass, damping and stiffness matrices and P is
the voltage-to-force transformation vector. Vector q represents the
plate response modal amplitudes and V is the applied voltage.
State-Space Equations
A model of a structure found via finite element or Rayliegh-Ritz
methods results in second-order differential equations of the form
[M]{9}+[C]{?}+[if]{?}={P}V (15)
Choosing state variables x^_ = q and = i, , equation 1 may be
reduced to a state-space representation as follows :
q = x =X2
/ / / (16)
q = X2= ~M-^Kq - M~'Cq + M~'PV.
Equations 15 can then be rewritten as
r 0 /I
0
1
1
1
1
A.
+
^2.
.^2_
[,] = [/ 0]
(17)
where [ ], { } are ignored. It should be noted that the vector q must be
multiplied by the shape fimctions to produce the actual displacement.
1223
Results
The model was used to investigate the response of a simply
supported plate. In order to maintain symmetry of the geometric
structure a piezoelectric actuator is assumed to be bonded to both the
top and bottom surfaces of the plate. So The symmetry of the elements
causes no extension of the plate midplane and the plate deforms in
pure bending. In this case the shape functions are assumed to be :
\{/(x,y) = 0
^(x,y) = 0 (18)
41 ( X, y ) = jinf x)sin( ^ y)
A-
where m and n are the number of half waves in the x and y directions.
The properties of the plate are given in Table 1 and its
dimensions are 4 = Q38m , 4, = Q30/w and 4=i.5876m/K. Tables 2 and 3
show the natural frequencies of the bare plate obtained by the thin
plate theory and the RR model, respectively. Since the shape functions
used in this example express the exact shape of the simply supported
plate, the natural frequencies included in Tables 2 and 3 are very close.
In order to excite the plate, a piezoelectric actuator with
configuration x, = 0.32 Xj = 0.36 m, = 0.04 OT and = 026 m is used above
and below. The natural frequencies of the plate after bonding the
piezoelectric actuator to the surface are given in Table 4. The results
show an increase in natural frequencies, showing that the added
stifness is more important than the added inertia.
Table 1 : Properties of the plate
207 7870 .292
Table 2 : Plate natural frequencies (rad / s) , Thin Plate Theory
1
2
3
4
1
437.5
1246.0
2593.5
4480.0
2
941.4
1749.9
3097.4
4983.9
3
1781.2
2589.7
3937.2
5823.7
4
2957.0
3765.5
5113.0
6999.5
1224
Table 3 : Plate natural frequencies (rad / s) , RR Model
/n
1
J
2
3
4
1
437.5
1245.9
2593.2
4479.1
2
941.3
1749.7
3097.0
4982.8
3
1781.1
2589.4
3936.6
5822.3
4
2956.7
3764.9
5111.9
6997.4
Table 4 : Plate-Piezo natural frequencies (rad / s)
/n
1
2
3
4
1
444.0
1257.1
2611.3
4502.8
2
957.2
1775.6
3182.1
5076.7
3
1854.6
2642.5
4072.3
6029.6
4
3076.6
3933.8
5224.6
7277.2
Initially, the piezoelectric actuator was excited by a constant DC
voltage. The result of this action is shown in figure 2 which shows the
dominant out of plane displacement around the location of the
piezoelectric actuator bonded to the surface of the plate. To show the
modes of vibration, the piezoelectric actuator was excited by a voltage
with frequencies near to the natural frequencies of modes (2,2) and
(1,3). The response in figures 3 , 4 show that the piezoelectric actuator
excited both of these modes.
Displacement Distribution (X-Y)
0 0
Figure 2. Static Displacement
1225
Displacement Distribution (X-Y)
0 0
Figure 3. Vibration of the plate, mode (2,2)
Displacement Distribution (X-Y)
0 0
Figure 4. Vibration of the plate, mode (3,1)
The frequency response of tihe plate at the centre was obtained
by exciting the piezoelectric actuator at a range of frequencies between
0 and 4000 rad/s. Figure 5 shows the frequency response of the plate
at its centre. The frequency response of the plate at y = 0.5L^ along the
x-direction is shown in figure 6. It can be seen that the amplitude of
vibration of some modes are very high compared to that of the other
modes. Special attention must be given for the suppression of vibration
of these modes.
1226
Frequency Response of the Plate
Figure 5. Frequency response of the plate at the centre
Frequency Response of the Plate
4000
Figure 6. Frequency response of the plate along x-direction
The results show that it is possible to predict the frequency
response of a plate when it is excited by a patch of piezoelectric
material. Consequently, a sensor model can be also added to the model
and a signal proportional to velocity fed back to the piezoelectric
patch. As a result, the closed loop frequency response of the plate can
be obtained theoretically which is very important for active vibration
control studies. This also permits the investigation of the optimal
1227
location of the actuators and the study of control algorithms for the
best possible vibration suppression before using any costly
experimental equipment.
For this purpose, two patches of piezoelectric (lOcmxScm and
5cmx4cm), whose specifications are listed in table 5, were bonded to
the surface of the plate in different locations and then the plate was
excited by a point force marked by "D" in the figure 7. In figure 7 the
dash lines are showing the nodal lines of a simply supported plate up
to mode (3, 3).
Table 5 : Properties of the actuator
(mm) EJxlO''^N/m')pJkg/m') dJxlO'^'mfv) ^
.2 6.25 7700 -180 .3
An actuator is most effective for control of a particular mode if
the sign of the strain due to the modal deflection shape is the same
over die entire actuator. Consequently, as can be seen from figure 7,
the actuators are placed between the nodal lines and at the points of
maximum curvature in order to obtain good damping effect on the
modes of interest. Then two accelerometers were located at the center
of the location of the actuators, marked by "S" in figure 7, in order to
have collocated sensor-actuators. The signals obtained by the
accelerometers are integrated and fed back to the actuators separately.
Therefore rate feedback was used in this configuration. This leads to
the feedback control law
V = kq
(19)
where k is an amplification factor or feedback gain. Substituting
equation 19 into equation 17 the closed-loop state-space representation
of the system can then be obtained as
0
-M-‘K
_ J
I —
1
M-^(C-kP)_
[^]=[/ 0]
(20)
1228
Then the effects of the actuators on vibration suppression were
investigated. At first, only the actuator near to the center of the plate
was used to suppress the vibration. The effect of this is shown in figure
8. As can be seen, damping in some modes are improved and some
modes are untouched.
The second actuator was added to the model to see its effect on
modes of vibration.
Figure 7. Plate with Bonded Piezoelectric Actuators
Frequency Response of the Plate
Figure 8. Open and Closed loop Frequency Response of the plate
solid line ; open loop, dash line : closed loop
1229
Frequency Response of the Plate
-50
_2qqI - 1 - 1 - 1 - 1 - 1 - 1 — — - 1 - 1
0 500 1000 1500 2000 2500 3000 3500 4000
Frequency (rad/s)
Figure 9. Open and Closed Loop Frequency Response of The Plate
solid line : open loop, dash line : closed loop
The open and closed loop frequency response of the plate when excited
by the point force and controlled by two actuators is shown in figure 9.
As can be seen, significant vibration suppression was obtained in both
lower and higher modes. Also, it shows that the place of actuators was
successfully chosen. This analysis showed that obtaining reasonable
but not necessarily optimal placement of actuators in structures is very
important in order to obtain a high level of damping in the modes of
interest. Obviously, bonding more than one piezoelectric actuator in
suitable locations helps to successfully suppress vibration of the plate.
Conclusions
A model of an active structure is fundamental to the design of
control strategies. It can be used to analyse the system and investigate
optimal control strategies without using costly experimental
equipment.
A Rayleigh-Ritz model has been developed to analyse the behaviour
of a thin plate excited by a patch of piezoelectric material. The model
has been used for a simply supported plate. It has been shown that the
model can predict natural frequencies of the plate alone very
accurately. The obtained mode shapes also correspond to the actual
mode shapes. The frequency response of the plate can be obtained to
show the suitability of the model for control design studies. This study
allowed the behaviour of the system in open and closed loop form for
active vibration control purposes to be investigated. Two piezoelectric
actuators were used to investigate their effectiveness on vibration
1230
suppression of the plate. The analysis showed that the location of two
actuators was very important to increase the level of damping in both
lower and higher frequency modes. The future work will be to extend
the model to analyse a plate with more than two patches of
piezoelectric material with optimal configuration, obtained by
controllability theory, and independent controller for vibration
suppression, and experimental verification of the theoretical analysis.
References
1. Crawley, E. F. and de Luis, J., Use of Piezoelectric Actuators as
Elements of Intelligent Structures, AlAA Journal, Vol. 25, No. 10,
1987, p. 1373.
2. Dosch, J., Leo, D. and Inman, D., Modelling and control for
Vibration Suppression of a Flexible Active structure, AIAA Journal
of Guidance, Control and Dynamics, Vol. 18, No. 2, 1995, p.340.
3. Bailey, T. and Hubbard J. E. Jr., Distributed Piezoelectric Polymer
Active Vibration Control of a Cantilever Beam, AIAA Journal of
Guidance, Control and Dynamics, Vol. 8, No. 4, 1985, p.605.
4. Dimitriadis, E. K., Fuller, C. R., Rogers C. A., Piezoelectric
Actuators for Distributed Vibration Excitation of Thin Plates,
Journal of Vibration and Acoustics, Vol. 113, No. 1, 1991, p. 100.
5. Azvine, B., Tomlinson, G. R. and Wynne, R. J., Use of Active
Constrained Layer Damping for Controlling Resonant, Journal of
Smart Materials and Structures, No. 4, 1995.
6. Rongong, J. A., Wright, J. R., Wynne, R. J. and Tomlinson, G. R.,
Modelling of a Hybrid Constrained Layer/Piezoceramic Approach
to Active Damping, Journal of Vibration and Acoustics, To appear.
7. Sadri, A. M., Wynne, R. J. and Cherry, A. S., Modelling and
Control of Active Damping for Vibration Suppression, UKACC
International Conference on Control' 96, 2-5 September 1996.
8. Bathe, K,, Finite Element Procedures in Engineering Analysis, Prentice-
Hall, Inc., 1982.
9. Blevins, R. D., Formulas for Natural Frequency and Mode Shapes, Van
Nostrand Remhold, NY 1979.
10. Thomson, W. T., Theory of Vibration with Applications, Prentice-Hall,
Inc., 1988.
1231
Active control of sound transmission into a rectangular
enclosure using both structural and acoustic actuators
S.M. Kim and MJ. Brennan
ISVR, University of Southampton, Highfieid, Southampton, S017 IBJ, UK
ABSTRACT
This paper presents an analytical investigation into the active control of sound
transmission in a ‘weakly coupled’ structural-acoustic system. The system
under consideration is a rectangular enclosure having one flexible plate
through which external noise is transmitted. Three active control systems
classified by the type of actuators are discussed. They are; i) a single force
actuator, ii) a single acoustic piston source, and iii) simultaneous use of both
the force actuator and the acoustic piston source. For all three control systems
the acoustic potential energy inside the enclosure is adopted as the cost
function to minimise, and perfect knowledge of the acoustic field is assumed.
The results obtained demonstrate that a single point force actuator is effective
in controlling well separated plate-controlled modes, whereas, a single
acoustic piston source is effective in controlling well separated cavity-
controlled modes provided the discrete actuators are properly located. Using
the hybrid approach with both structural and acoustic actuators, improved
control effects on the plate vibration together with a further reduction in
transmitted noise and reduced control effort can be achieved. Because the
acoustic behaviour is governed by both plate and cavity-controlled modes in a
‘weakly coupled’ structural-acoustic system, the hybrid approach is desirable
in this system.
1. INTRODUCTION
Analytical studies of vibro-acoustic systems have been conducted by many
investigators to achieve physical insight so that effective active control
systems can be designed. It is well established that a single point force actuator
and a single acoustic piston source can be used to control well separated
vibration modes in structures and well separated acoustic modes in cavities,
respectively, provided that the actuators are positioned to excite these
modes [1,2]. Active control is also applied to structural-acoustic coupled
systems for example, the control of sound radiation from a piate[3-6] and the
sound transmission into a rectangular enclosure[7-8]. Meirovitch and
1233
Thangjitham[6], who discussed the active control of sound radiation from a
plate, concluded that more control actuators resulted in better control effects.
Pan et al[Z] used a point force actuator to control sound transmission into an
enclosure, and discussed the control mechanism in terms of plate and cavity-
controlled modes.
This paper is concerned with the active control of sound transmission into a
‘weakly coupled’ structural-acoustic system using both structural and acoustic
actuators. After a general formulation of active control theory for structural-
acoustic coupled systems, it is applied to a rectangular enclosure having one
flexible plate through which external noise is transmitted. Three active control
systems classified by the type of actuators are compared using computer
simulations. They are; i) a single force actuator, ii) a single acoustic piston
source, and iii) simultaneous use of both the force actuator and the acoustic
piston source. For all three control systems the acoustic potential energy inside
the enclosure is adopted as the cost function to minimise, and perfect
knowledge of the acoustic field is assumed. The effects of each system are
discussed and compared, and a desirable control system is suggested.
2. THEORY
2.1 Assumptions and co-ordinate systems
Consider an arbitrary shaped enclosure surrounded by a flexible structure and a
acoustically rigid wall as shown in Figure 1. A plane wave is assumed to be
incident on the flexible structure, and wave interference outside the enclosure
between the incident and radiated waves by structural vibration is neglected.
Three separate sets of co-ordinates systems are used; Co-ordinate x is used for
the acoustic field in the cavity, co-ordinate y is used for the vibration of the
structure, and co-ordinate r is used for the sound field outside the enclosure.
The cavity acoustic field and the flexible structure are governed by the linear
Helmholtz equation and the isotropic thin plate theory[9], respectively. The
sign of the force distribution function and normal vibration velocity are set to
be positive when they direct inward to the cavity so that the structural
contribution to acoustic pressure has the same sign as the acoustic source
contribution to acoustic pressure.
Weak coupling rather than full coupling is assumed between the structural
vibration system and the cavity acoustic system. Thus, the acoustic reaction
force on the strucural vibration under structural excitation and the structural
induced source effect on the cavity acoustic field under acoustic excitation is
neglected. This assumption is generally accepted when the enclosure consists
of a heavy structure and a big volume cavity. It is also assumed that the
coupled response of the system can be described by finites summations of the
1234
uncoupled acoustic and structural modes. The uncoupled modes are the rigid-
walled acoustic modes of the cavity and the in vacuo structural modes of the
structure. The acoustic pressure and structural vibration velocity normal to the
vibrating surface are chosen to represent the responses of the coupled system.
2.2 Structural-acoustic coupled response
The acoustic potential energy in the cavity is adopted as the cost function for
the global sound control, which is given by [2]
where, and Co respectively denote the density and the speed of sound in air,
and /?(x,ft)) is the sound pressure inside the enclosure.
The vibrational kinetic energy of the flexible structure, which will be used to
judge the control effect on structural vibration, is given by[l]
where, p., is the density of the plate material, h is the thickness of the plate.
If the acoustic pressure and the structural vibration are assumed to be
described by a summation of N and M modes, respectively, then the acoustic
pressure at position x inside the enclosure and the structural vibration velocity
at position y are given by
N
P(X,(B) =
n=I
M
ni=l
where, the N length column vectors ^ and a consist of the array of uncoupled
acoustic mode shape functions and the complex amplitude of the
acoustic pressure modes a^^(co) respectively. Likewise the M length column
vectors O and b consist of the array of uncoupled vibration mode shape
functions 0n,(y) and the complex amplitude of the vibration velocity modes
respectively.
The mode shape functions \f/^{x) and (l)Jy) satisfy the orthogonal property
in each uncoupled system, and can be normalised as follows.
V = lwlMdV (5)
S,=lfyy)dS (6)
1235
where, V and S/ are the volume of the enclosure and the area of the flexible
structure, respectively. Since mode shape functions are normalised as given by
(Eq. 5), the acoustic potential energy can be written as
£ =_L^a“a (7)
' 4p„c^
Similarly from (Eq. 2) and (Eq. 6), the vibrational kinetic energy can be
written as
£ =££^b«b (8)
2
Where superscript H denotes the Hermitian transpose.
For the global control of sound transmission, it is required to have knowledge
of the complex amplitude of acoustic pressure vector a for various excitations.
The complex amplitude of the n-th acoustic mode under structural and
acoustic excitation is given by[9, 10]
^1. (^) = 4 (^)(^ (x)‘y(x, co)dv + i//'„ (yMy, j (9)
where, s(x,Ci)) denotes the acoustic source strength density function in the
cavity volume V”, and w(y,6)) denotes the normal velocity of the surrounding
flexible structure on surface 5/-. The two integrals inside the brackets represent
the nth acoustic modal source strength contributed from s(x,Q)) and u(y,co) ,
respectively. The acoustic mode resonance term An(fi)) is given by
A,.(co) = - - - (10)
o}--Q)-+j2C„<a„(o
where a)„and C„ are the natural frequency and damping ratio of the nth
acoustic mode, respectively.
Substituting (Eq. 4) into (Eq. 9) and introducing the modal source strength
q,: = j ¥n X. 0))dV , then we get
2 / M
a„ (®) = A, (®) (®) + S O',,™ • b,„((0)
y V Hi=i
where, C„,,„ represents the geometric coupling relationship between the
uncoupled structural and acoustic mode shape functions on the surface of the
vibrating structure Sf and is given by[l 1]
c,,,„ = lvJy)<t>Jy’0>)ds (12)
If we use L independent acoustic control sources, can be written as
/=! \l /=!
1236
where = - j W ^ control source strength q,^i{co)
having an area of Scjj is defined at Xc,i.
Thus, the complex amplitude of acoustic modal pressure vector a can be
expressed as
a = Z.(D,q,+Cb) (14)
2
whereZa= -^A.
The matrix A is a (NxN) diagonal matrix in which each (n,n) diagonal term
consists of A„, the (NxM) matrix C is the structural-acoustic mode shape
coupling matrix, the (NxL) matrix Dq determines coupling between the L
acoustic source locations and the N acoustic modes, the L length vector (jc is
the complex strength vector of acoustic control sources, and b is the complex
vibrational modal amplitude vector. The {NxN) diagonal matrix Za can be
defined as the uncoupled acoustic modal impedance matrix which determines
the relationship between the acoustic source excitation and the resultant
acoustic pressure in modal co-ordinates of the uncoupled acoustic system.
Generally the impedance matrix is symmetric but non-diagonal in physical co¬
ordinates, however the uncoupled modal impedance matrix is diagonal
because of the orthogonal property of uncoupled modes in modal co-ordinates.
Since the flexible structure in Figure 1 is assumed to be governed by the
isotropic thin plate theory, the complex vibrational velocity amplitude of the
mth mode can be expressed as[10]
= + (15)
p,hSf ^
where, again p.v is the density of the plate material, h is the thickness of the
plate, is the area of flexible structure. Inside the integral /(y,ty) , p""\'^.(0) ,
and p‘"'(y,ty) denote the force distribution function, and the exterior and
interior sides of acoustic pressure distribution on the surface 5/ , respectively.
Because of the sign convention used, there is a minus sign in front of
(y, (O) . The structural mode resonance term Bm((0) can be expressed as
-(0^+j2^„^0),„co
where co^ and Cm natural frequency and the damping ratio of m-th
mode, respectively. Substituting (Eq. 3) into (Eq. 15), then we get
b,niO})=——B,„ico)\ g,,,„ico)+ g^,„i(o)-'£cl„-a,(co) (17)
p,hSf \ „=i ;
1237
where, L = L
• JSj- •'■i/
and Cl^ = C,„.. ■
If we use K independent point force actuators, the m-th mode generalised force
due to control forces, gc.m, can be written by
= XI <P,n(y)S(y-ycjc)dsf,j,ico) = X^/.».t/<.-.t(®)
k=\ ^ k=]
(18)
where 0,„(y)5(y-y.,t)rfS, and the k-th control point force l,{co)
JSj
is located at y^- it.
Thus the modal vibrational amplitude vector b can be expressed as
b = Y,(g,+Dff,-C’'a) (19)
where Ys = — ^ — B.
P..hSjr
The matrix B is a (MxM) diagonal matrix in which each (m,m) diagonal term
consists of B,„, is the transpose matrix of C, the (NxK) matrix Df
determines coupling between the K point force locations and the M structural
modes, gp is the generalised modal force vector due to the primary plane wave
excitation, the K length vector fc is the complex vector of structural control
point forces, and a is the complex acoustic modal amplitude vector. The
(MxM) diagonal matrix Ys can be defined as the uncoupled structural modal
mobility matrix which determines the relationship between structural
excitation and the resultant structural velocity response in modal co-ordinates
of the uncoupled structural system. As with the uncoupled acoustic impedance
matrix Za, note that Ys is a diagonal matrix.
From (Eq. 14) and (Eq. 19), we get
a = (l + Z,CY,C’^)''z,(D,q,+CY,g,+CWc)
(20)
b = (I + Y,C’'Z,C)'' Y,(gp + Dff, - C^Z.D^q,)
(21)
Since weakly coupling is assumed i.e. = 0 and Y^C^Z^C = 0 ,
then
we get
a = Z,(D,q,+CY,g,+CY,D,f.)
(22)
b = X(g,+D,f,-C%D,q,)
(23)
Although the formulation developed above covers fully coupled systems, weak
coupling is assumed hereafter for the convenience of analysis.
1238
In order to minimise the sound transmission into the cavity, two kinds of
actuators are used: a single point force actuator for controlling the structural
vibration of the plate and a single rectangular type acoustic piston source for
controlling the cavity acoustic pressure. The rectangular piston source is
centred at (1.85,0.15,0) with the area of 0.15m by 0.15m, This location was
chosen because the sound pressure of each mode in a rectangular cavity is a
maximum at the corners, and thus the control source is placed away from the
acoustic nodal planes [2]. For a similar reason, the point force actuator is
located at (9/20Li, L2/2) on the plate, at which there are no nodal lines within
the frequency range of interest. Table 2 shows the natural frequencies of each
uncoupled systems and their geometric mode shape coupling coefficients
which are normalised by their maximum value. Some of natural frequencies
which are not excited by the given incident angie((p = 0°) were omitted. The
(m/, m2) and («/, 112, ns) indicate the indices of the m-th plate mode and the n\h
cavity mode, and corresponding the uncoupled natural frequencies of the plate
and the cavity are listed. A total 15 structural and 10 acoustic modes were used
for the analysis under 300 Hz, and no significant difference was noticed in
simulations with more modes.
3.2 Active minimisation of the acoustic potential energy
This section considers an analytical investigation into the active control of the
sound transmission into the rectangular enclosure in Figure 2. Three active
control strategies classified by the type of actuators are considered. They are;
i) a single force actuator, ii) a single acoustic piston source, and iii)
simultaneous use of both the force actuator and the acoustic piston source.
Although the formulation developed in this paper is not restricted to a single
actuator, each single actuator was used to simplify problems so that the control
mechanisms could be understood and effective guidelines for practical
implementation could be established.
3.2.1 control using a single force actuator
A point force actuator indicated in Figure 2 is used as a structural actuator and
the optimal control strength of the point force actuator can be calculated using
(Eq. 26). Figure 3(a) shows the acoustic potential energy of the cavity with
and without the control force. To show how this control system affects the
vibration of the plate, the vibration kinetic energy of the plate obtained from
(Eq. 8) is also plotted in Figure 3(b). On each graph, natural frequencies of the
plate and the cavity are marked and ‘o’ at the frequencies, respectively. It
can be seen that the acoustic response of uncontrolled state has peaks at both
1239
plate and cavity resonances, and the vibration response of uncontrolled state is
governed by the plate resonances only because of ‘weak coupling’.
Examining Figure 3(a,b) it can be seen that at the 1st, 2nd, 4th, and 5th plate
modes corresponding to 52 Hz, 64 Hz, 115 Hz, and 154 Hz, respectively there
is a large reduction of the acoustic potential energy. This is because the sound
field at these frequencies is governed by the plate vibration modes, and a
single structural actuator is able to control the corresponding vibration mode to
minimise sound transmission.
The structural actuator reduces sound at cavity-controlled modes as well(
especially the 2nd and 3rd cavity modes corresponding respectively to 85 Hz
and 170 Hz), however it has to increase plate vibration significantly. It shows
that minimisation of the acoustic potential energy does not always bring the
reduction of structural vibration, and vice versa. Since a cavity-controlled
mode is generally well coupled with several structural modes, a single
structural actuator is not able to deal with several vibration modes because of
'control spillover\A\. This is the reason why a single acoustic piston source
used in the next section was introduced. However, it is clear that a single
point force actuator is effective in controlling a well separated plate-controlled
mode provided the actuator is not located close to the nodal line.
3.2.2 control using a single piston force source
A single acoustic piston source indicated in Figure 2 is used for controlling
the acoustic sound field directly. The optimal control source strength of the
acoustic piston source can be determined using (Eq. 26). Figure 4 shows the
acoustic potential energy of the cavity and the vibrational kinetic energy of the
plate with and without the control actuators.
Since a plate-controlled mode is generally coupled with several cavity modes,
the control effect of the acoustic source is not effective at plate-controlled
modes (e.g. 52 Hz, 64 Hz, 115 Hz etc.). Whereas, it is more able to reduce
transmitted sound at the cavity-controlled modes ( e.g. 0 Hz, 85 Hz, 170 Hz
and 189 Hz) than the structural actuator. As can be noticed from (Eq. 22), the
external incident wave and force excitation have the same sound transmission
mechanism, which is vibrating the plate and transmitting sound through the
geometric mode shape coupling matrix C. Thus, it can be said that the
structural actuator is generally effective in controlling sound transmission. At
cavity controlled modes, however, several vibration modes are coupled with
an acoustic mode. It means that a single acoustic source is more effective than
a single structural actuator since a single actuator is generally able to control
only one mode. From the results, it is clear that a single acoustic piston source
is effective in controlling well separated cavity-controlled modes. It is
interesting that there is not much difference in the vibrational kinetic energy
1240
with and without control state. It means that the acoustic actuator is able to
reduce sound field globally without increasing plate vibration.
3.2.3 control using both the piston source and the structural actuator
In this section, a hybrid approach, simultaneous use of both the point force
actuator and the acoustic piston source, is applied. The optimal strength of the
force actuator and the piston source can be obtained from Eq. (26). Figure 5
shows the acoustic potential energy of the cavity and the vibrational kinetic
energy of the plate with and without the control actuators. Even at the cavity-
controlled modes, it can be seen that a large reduction in the acoustic potential
energy is achieved without significantly increasing the structural vibration. In
general, more control actuators result in better control effects[6]. However, the
hybrid approach with both structural and acoustic actuators in the system does
not merely mean an increase in the number of actuators. As demonstrated in
the last two sections, a single structural actuator is effective in controlling
well separated plate-controlled modes and a single acoustic actuator is
effective for controlling well separated cavity-controlled modes. Since the
acoustic response is governed by both plate-controlled and cavity-controlled
modes, the hybrid control approach can be desirable for controlling sound
transmission in a ‘weakly-coupled’ structural acoustic system.
To investigate the control efforts of each control system, the amplitude of the
force actuator and the source strength of the piston source are plotted in
Figure 6. There is a large decrease of the force amplitude at the well
separated cavity-controlled modes, e.g. 85 Hz and 170 Hz, by using the both
actuators. This trend can also be seen in the case of the piston source strength,
especially at the 1st and 2nd structural natural frequency (52 Hz, 64 Hz). By
using the hybrid approach, simultaneous use of both actuators, better control
effects on the vibration of the plate, the transmission noise reduction and the
control efforts of the actuators can be achieved.
4. Conclusion
The active control of the sound transmission into a ‘weakly coupled’
structural-acoustic system has been considered. The results obtained
demonstrates that a single point force actuator is effective in controlling well
separated plate-controlled modes, whereas, a single acoustic piston source is
effective in controlling well separated cavity-controlled modes.
By using the hybrid approach with both structural and acoustic actuators,
improved control effects on the plate vibration, further reduction in sound
1241
transmission, and reduced control efforts of the actuators can be achieved.
Since the acoustic behaviour is governed by both plate and cavity resonances,
the hybrid control approach can be desirable in controlling sound transmission
in a ‘weakly coupled’ structural-acoustic system.
References
1. C.R. FULLER, S.J. ELLIOTT and P.A. NELSONActive control of
vibration, Academic Press Limited, 1996
2. P.A. NELSON and S.J. ELLIOTT Active control of sound, Academic
Press Limited, 1992
3. B.-T. WANG, C.R. FULLER and K. DIMITRIADIS Active control of
noise transmission through rectangular plates using multiple
piezoelectric or point force actuators Journal of the Acoustical Society
of America, 1991, 90(5), 2820-2830.
4. M.E. JOHNSON and S. J. ELLIOTT Active control of sound radiation
using volume velocity cancellation. Journal of the Acoustical Society of
America, 1995, 98(4), 2174-2186.
5. C.R. FULLER, C.H. HANSEN and S.D. SNYDER Active control of
sound radiation from a vibrating rectangular panel by sound sources
and vibration inputs: an experimental comparison, Journal of Sound
and Vibration, 1991, 145(2), 195-215.
6. L. MEIROVrrCH and S. THANGJITHAM Active control of sound
radiation pressure, Trans, of the ASMS Journal of Vibration and
Acoustics, 1990, 112, 237-244.
7. S.D. SNYDER and N. TANAKA On feedforward active control of
sound and vibration using vibration error signals, Journal of the
Acoustical Society of America, 1993, 94(4), 2181-2193.
8. J. PAN C.H. HANSEN and D. A. BIES Active control of noise
transmission through a panel into a cavity : I. analytical study. Journal
of the Acoustical Society of America, 1990, 87(5), 2098-2108.
9. P.M. MORSE and K.U. INGARD Theoretical Acoustics, McGraw-
Hill, 1968
10. E.H. DOWELL, G.F. GORMAN HI, and D.A. SMITH
Acoustoelasticity : general theory, acoustic modes and forced response
to sinusoidal excitation, including comparisons with experiment.
Journal of Sound and Vibration, 1977, 52(4), 519-542.
11. F. FAHY Sound and Structural Vibration, Radiation, Transmission
and Response, Academic Press Limited, 1985
1242
incident plane wave
Figure 1 A structural acoustic coupled system with the volume V and its flexible boundary
surface S/.
Figure 2 The rectangular enclosure with one simply supported plate on the surface Sf on
which external plane wave is incident with the angles of (cp = 0°) and (0=45°).
Table 1 Material properties
Material
Density
Phase speed
Young’s
Poisson’s
Damping
(kg/m^)
(m/s)
modulus (N/m^)
ratio (v)
ratio (0
Air
1.21
340
-
-
0.01
Steel
7870
-
207x10®
0.292
0.01
1243
Table 2 The natural frequencies and geometric mode shape coupling coefficients of each
uncoupled system
Order:
Plate,
1
2
3
4
■„ , 5.
7
10
■c
’Type
(1,1)
(2,1)::
0,1) :
(4,1)
(5,1)
(6,1)
(7,1)
Cavity^
' Freq.-
52 Hz
64 Hz
86Hz
115 Hz
154 Hz
200 Hz
;256Hz
1
(0,0,0)
0 Hz
0.71
0
0.24
0
0.14
0
O.IO
2
(1,0.0)
85 Hz
0
0.67
0
0.27
0
0.17
0
3
(2,0.0)
170 Hz
-0.33
0
0.60
0
0.24"
0
0.16
4
(0,0.1)
189 Hz
-1.00
0
-0.33
0
-0.20"
0
-0.14
5
(1,0,1)
207.HZ
0
-0.94
0
-0.38
0
-0.24
0
6
(2;0,1)
254 Hz
0.47
0
-0.85
0
-0.34
0
-0.22
7
(3,0,0)
255 Hz
0
-0.40
0
0.57
0
0.22
0
1 oo ISO 200 2SO
Frequency
(a) the acoustic potential energy of the cavity
TSO
Frequency
(b) the vibrational kinetic energy of the plate(dB ref .= 10 ^ J)
Figure 3 Effects of minimising the acoustic potential energy using a point force actuator (
solid line : without control, dashed line : with control ), where **’ and ‘o’ are at uncoupled
plate and cavity natural frequencies, respectively.
301
Frequency
(a) the acoustic potential energy of the cavity
Figure 4 Effects of minimising the acoustic potential energy using an acoustic piston source
( solid line ; without control, dashed line : with control ), where and ‘o’ are at uncoupled
plate and cavity natural frequencies, respectively.
Figure 5 Effects of minimising the acoustic potential energy using both a point force
actuator and an acoustic piston source - continued
2
Frequency
(b) the vibrational kinetic energy of the pIate(dB ref =10‘^ J)
Figure 5 Effects of minimising the acoustic potential energy using both a point force
actuator and an acoustic piston source ( solid line : without control, dashed line: with control
), where and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively.
Frequency
(a) the strength of the force actuator
Frequency
(b) the strength of the piston source( unit: mVsec)
Figure 6 Comparison of control efforts of the three control strategies; using each actuator
separately ( solid line ) and using both the force actuator and the piston source (dashed line )
, where and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively.
1246
A DISTRIBUTED ACTUATOR FOR THE
ACTIVE CONTROL OF SOUND
TRANSMISSION THROUGH A PARTITION
TJ. Sutton, M.E. Johnson and S.J. Elliott
Institute of Sound and Vibration Research
University of Southampton, Southampton S017 IBJ
ABSTRACT
The paper considers the problem of active control of soimd transmission
through a partition using a single distributed actuator. The use of shaped,
distributed actuators rather than point sources or locally-acting piezoceramic
elements offers the possibility of controlling the volume velocity of a plate
without giving rise to control spillover and avoids an increase in the sound
radiated by uncontrolled structural modes. Specifically, a form of distributed
piezoelectric actuator is described in which the electrode takes the form of a
set of quadratic strips and serves to apply a roughly uniform normal force
over its surface.
INTRODUCTION
The strong piezoelectric properties of the polymer polyvinylidene fluoride
(PVDF) were discovered in 1969 [1]. The material is lightweight, flexible,
inexpensive and can be integrated into engineering structures for strain
sensing and to apply distributed forces and moments for the active control of
vibration and sound transmission. Such ^smarf materials offer the possibility
of providing lightweight sound-insulating barriers for application to aircraft,
ground-based transport and in buildings.
Lee [2] has set out the underlying theory of active laminated structures in
which one or more layers of flexible piezoelectric material are attached to a
plate. Practical sensors using PVDF material have been implemented by
Clark and Fuller [3], Johnson and Elliott [4-6], and others. In these cases thin
PVDF films were attached to the structure to sense integrated strain over a
defined area. In [4] for example a distributed sensor was developed whose
output is proportional to the integrated volume velocity over the surface of a
plate.
1247
A number of studies have been carried out in which distributed piezoelectric
actuators form a layer of a laminated system. In [7] the shape of a distributed
piezoelectric actuator was chosen to be orthogonal to all but one of the natural
modeshapes of the cylindrical shell system xmder control. Using this
approach a set of actuators could be matched to the modes of the system
under control, avoiding control spillover {i.e. the excitation of tmcontrolled
structural modes).
In the present paper a single shaped PVDF actuator is applied to a thin plate
to control the noise transmission through it. The shape of the actuator is
chosen specifically to apply an approximately uniform force to the plate.
Such an actuator can be used to cancel the total volume velocity of the plate
and therefore substantially to reduce the radiated sound power. (If volume
velocity is measured at the plate surface there is no requirement for a remote
error microphone.) As noted by Johnson and Elliott [6], the soimd power
radiated by a plate which is small compared with an acoustic wavelength
depends mainly on the volume velocity of the plate. The simulations in [6]
show that provided the plate is no larger than half an acoustic wavelength, a
single actuator used to cancel volume velocity will achieve similar results to a
strategy in which radiated power is minimised. It is possible to envisage a
large partition made up of a number of active plate elements designed on this
basis.
CALCULATION OF NORMAL FORCES IN THE PLATE
In this section the equation of motion of the plate and attached PVDF layers is
set out. The film thickness is assumed to be 0.5 mm. The analysis broadly
follows that of Dimitriadis, Fuller and Rogers [8], but the individual
piezoelectric coefficients and included separately as is appropriate
for PVDF and a sensitivity function is included to account for variations in
electrode shape. In addition the bending stiffness of the piezoelectric film is
included (it is not negligible as the whole plate is covered). The nomenclature
matches that used by Fuller, Elliott and Nelson [9] but here the analysis leads
to the inhomogeneous wave equation for the plate-actuator system.
We consider an aluminium plate of thickness Ih^ as shown in Figure 1. The
plate is covered on its upper and lower surfaces with a piezoelectric film of
sensitivity (}>(x,y)d-p in which d.^ is the strain/ electric field matrix of the material
(3x6 array) and <^{x,y) is a spatially-varying sensitivity function
(0 < (j)(x,y) < 1 ). The two piezoelectric films are assumed to be identical but
the same drive voltage is applied with opposite polarity to the lower film. As
a result of this antisymmetric arrangement, the plate is subject to pure
bending with no straining of the plate midplane.
1248
In line with other similar calculations [2,9] it is assumed that any line
perpendicular to the midplane before deformation will remain perpendicular
to it when the plate /PVDF assembly is deformed. As a result, the strain at
any point in the assembly is proportional to distance z through it. (z=0 is
defined to be on the midplane of the plate as indicated in Figure 1.) The
direct and shear strains throughout the whole assembly (e^, Ey, E^y) are then
given by [10]:
3^^
d^w
(1-3)
in which w is the displacement of the midplane in the z direction.
The corresponding stresses in the plate (only) follow from Hooke's law as in
the standard development for thin plates:
p _ _ p_
=
i-v:
A ^ p
0'' = — — -
2a+vj
(4-6)
in which is the Young's modulus and Vp is Poisson's ratio for the plate
material. Stress in the piezoelectric film follows from the constitutive
equations for the material [2]. The direct and shear stresses for the upper
piezo film are designated of of and respectively:
's/
K
or"’
= C ^
1 -
CO
_^36.
in which is the voltage applied across the actuator fibn (thickness /rj and
4 and are the strain/field coefficients for the material. For PVDF d^, =
0, but it is included in the analysis for completeness. As in [2], the stiffness
matrix C is given by:
1249
0
c
^pe ^ pe^pe
1 —
^ ^ pe ^ pe
^ pe^pe ^pe
1-v' 1-v^
0 0
0
^pe
2(l + v^J
(8)
The stresses in the lower piezoelectric layer are designated and
, and the form of the expression is similar to the upper layer except that
the voltage is applied with reversed polarity:
's/
^31
V
= c
+ (^{x,y)
a
Summing moments about the x and y axes for a small element dxby of the
plate yields the moment per unit length about the y-axis and My about
the x-axis; also the corresponding twisting moments per imit length, My^ and
M^=jc!’’zdz+
-III, ~K-K ^‘b
(10)
h,, -III, ‘‘b+lh,
My = j a^yZdz + J + J cf'^zdz
-III, -hh-K h
(11)
h,, -III, l>h+lh,
= |<t;,z*+ ja^fhdz
(12)
-III, -K-K h
and My^=M,y.
The vertical acceleration at each point of the plate d^w/dt^ is obtained by
taking moments about the x and y axes for a small element 6x6y and
resolving vertical forces as in standard thin plate theory. If the plate is acted
on by some external force per unit area p(x,y)f(t) then the vertical motion of
the plate is described by:
dx^ dxdy dy^ dt^
= -p(x,y)f(t)
(13)
1250
in which m is the mass/ area of the plate-film assembly.
The equation of motion of the plate complete with attached piezoelectric film
is obtained by combining the above equations. For convenience the following
constants are defined;
n _
'' 3(1
(bending stiffness of plate, thickness 2/i,) (14)
3(1 -V^)
(bending stiffness due to PVDF) (15)
+2/1,)
The equation of motion of the plate assembly including upper and lower
piezoelectric layers is then given by:
3^(l)(^,y)
3>(j:,y)
The left-hand side of this equation determines the free response of the plate-
film assembly and is recognised as the standard form for a thin plate. The
first term on the right-hand side is the assumed externally applied normal
force per imit area. The second term on the right-hand side gives the effective
normal force per unit area applied to the plate due to the two piezoelectric
films driven by a voltage . It is clear that this force depends on the
spatially-varying sensitivity which has been assumed for the piezoelectric
material.
An examination of Eqns. (16) and (17) shows that the normal force applied by
the piezoelectric film depends on the sum of the plate thickness and the
thickness of one of the film layers. If the film is much thinner than the plate
(h^ « hy) then the applied force becomes independent of the film thickness and
depends only on the plate thickness, the applied voltage, the electrode pattern
and the material constants. Eqn. (17) also shows that the normal force is
applied locally at all points on the plate. No integration is involved, and so in
contrast with a volume velocity sensor designed using quadratic strips [5], the
force does not depend on the plate boundary conditions in any way.
Furthermore, there is no need to use two films oriented at 90 degrees to cancel
1251
the cross-sensitivity It is also worth noting that no assumption has been
made about the modeshapes on the plate.
We can create a uniform force actuator by choosing:
^ = constant, and (18)
Bx
(19)
This can approximately be achieved by depositing electrodes in the form of
narrow strips whose width varies quadratically in the x-direction. (Note that
the x-direction is defined as the direction of rolling of the PVDF material, Le.
the direction of for maximum sensitivity.) The form of the electrodes is as
shown in Figure 2. With this pattern the sensitivity function takes the form:
(t)(x,>') = (20)
where is the length of the strip. Thus (t)(-^>^) - ^ at x = 0 and x = (no
electrode), while (|)(x, 31) = 1 halfway along at x = LJ2 (electrode fully covers
the film).
APPLICATION TO A THIN ALUMINIUM PLATE
If a plane wave of sound pressure level 94 dB (say) is normally incident on a
hard surface, it will exert a pressure of 2 Pa rms on that surface. If this
incident pressure is counterbalanced by a uniform force actuator applied to a
plate, then the plate could in principle be brought to rest. Thus for active
control of everyday noise levels the uniform force actuator will need to be
able to generate a normal force /area of a few pascal over the surface of the
plate. (When the incident wave impinges on the plate at an oblique angle,
many natural modes of the plate will be excited and it will not be possible to
bring it perfectly to rest with a single actuator; however it will remain
possible to cancel the plate volume velocity as explained earlier.)
By way of example an aluminium plate of thickness 1 mm will be assumed,
with a free surface measuring 300 x 400 mm. Attached to each side is a PVDF
film of thickness 0.5 mm. One electrode of each panel would be masked to
give quadratic strips of length 300 mm as shown in Figure 2. (The width of the
strips is unimportant, but should be significantly smaller than the structural
wavelength of modes of interest on the plate.) In this case it turns out that
= 6.64 for the plate, and
1252
Dp, = 1.27 for the PVDF film.
The piezoelectric constants for the film are typically
4 = 23 X m/V and
4 = 3 X lO’"' m/V, giving
Cp, = 3.28x10^
Finally the double derivative of the sensitivity function turns out to be
= 88.9 (300 mm strip length)
dx^
The bending stiffness of the 1 mm aluminium plate is increased by 20% due to
the addition of two layers of PVDF film of thickness 0.5 mm each. The force
per unit area due to the actuator is obtained from Eqn. (17):
force/area = 4,(^3, +Vp,4)|^'^3
= 6.96x10’^ V3 Pa
Thus 1000 volt rms would yield a tmiform force/area of close to 7 Pa. This is
not an impracticable voltage level, but previous experience at ISVR suggests
that care would need to be taken to avoid electrical breakdown through the
air between electrodes, or over damp surfaces.
ACTIVE CONTROL OF SOUND TRANSMISSION
In reference [6] Johnson and Elliott have presented simulations of the active
control of harmonic sound transmitted through a plate using a uniform force
actuator. Their actuator might be realised along the lines described in this
paper. In the simulatioi\s presented in [6] the uniform force actuator is used
with a matched volume velocity sensor having the same electrode shape [4].
The advantage of this configuration is that the actuator can be used to drive
the net volume velocity of the plate to zero without exciting high order
structural modes in the process (control spillover). Simulations of a
300x380x1 mm aluminium plate showed that reductions in transmitted sound
power of around 10 dB were achievable in principle up to 600 Hz using this
matched actuator-sensor arrangement.
A further advantage of the distributed matched actuator-sensor pair is that
the secondary path through the plate (for active control) is minimum phase
[6], giving good stability characteristics if a feedback control loop is
implemented to control random incident soimd for which no reference signal
is available.
1253
CONCLUSION
A design of distributed piezoelectric actuator has been presented which
generates a roughly uniform force over the surface of a plate. An example
calculation shows that the design is capable of controlling realistic soimd
pressure levels. When used in combination with a matched volume velocity
sensor, the actuator-sensor pair will have minimum-phase characteristics and
will offer the possiblity of feedback control in which neither a reference signal
nor a remote error sensor will be required.
ACKNOWLEDGEMENT
The financial support of the European Community under the Framework IV
programme is gratefully acknowledged. (Project reference: BRPR-CT96-0154)
REFERENCES
[1] G.M. Sessler (1981) JAcoust Soc Am 70(6) Dec 1981 1596-1608
Piezoelectricity in polyvinylidene fluoride
[2] C.K. Lee (1990) JAcoust Soc Am 87(3) Mar 1990 1144-1158 Theory of
laminated piezoelectric plates for the design of distributed sensors/ actuators.
Part I: Governing equations and reciprocal relationships
[3] R.L. Clark and C.R. Fuller (1992) JAcoust Soc Am 91(6) June 1992 3321-3329
Modal sensing of efficient acoustic radiators with polyvinylidene fluoride
distributed sensors in active structural acoustic control approaches
[4] M.E. Johnson, S.J. Elliott and J.A. Rex (1993) ISVK Technical Memorandum
723. Volume Velocity Sensors for Active Control of Acoustic Radiation
[5] M.E. Johnson and S.J. Elliott (1995) Proceedings of the Conference on Smart
Structures and Materials 27 Feb-3 Mar 1995, San Diego, Calif. SPIE Vol 2443.
Experiments on the active control of sound radiation using a volume velocity
sensor
[6] M.E. Johnson and S.J. Elliott (1995) JAcoust Soc Am 98(4) Oct 1995 2174-
2186. Active control of sound radiation using volume velocity cancellation
[7] H.S. Tzou, J.P. Zhong and J.J. Hollkamp (1994) Journal of Sound and
Vibration 177(3) 363-378 Spatially distributed orthogonal piezoelectric shell
actuators: theory and applications
[8] E.K. Dimitriadis, C.R. Fuller and C.A. Rogers (1991) Transactions of the
ASME, Journal of Vibration and Acoustics 113 100-107 Piezoelectric actuators for
distributed vibration excitation of thin plates
[9] C.R. Fuller, S.J. Elliott and P.A. Nelson (1996) Active Control of Vibration.
Academic Press, London.
[10] G.B. Warburton (1976) The Dynamical Behaviour of Structures, 2nd Edition.
Pergamon Press, Oxford.
1254
layers of
PVDF film
Figure 1 : Schematic diagram of thin plate
covered on both sides with a layer of PVDF film
Figure 2: Electrode pattern of quadratic strips for
uniform-force actuator
1255
1256
CONTROL OF SOUND RADIATION FROM A FLUID-LOADED PLATE
USING ACTIVE CONSTRAINING LAYER DAMPING
J. Ro, A. Al-Ali and A. Baz
Mechanical Engineering Department
The Catholic University of America
Washington D. C. 20064
Abstract
Sound radiation from a vibrating flat plate, with one side subjected to
fluid-loading, is controlled using patches of Active Constrained Layer Damping
(ACLD). The fluid-structure-controller interaction is modeled using the finite
element method. The damping characteristics of the ACLD/plate/fluid system are
determined and compared with the damping characteristics of plate/fluid system
controlled with conventional Active Control (AC) and/or Passive Constrained
Layer Damping (PCLD) treatments. Such comparisons are essential in
quantifying the individual contribution of the active and passive damping
components to the overall damping characteristics, when each operates separately
and when both are combined to interact in unison as in the ACLD treatments.
I. INTRODUCTION
When a structure is in contact with or immersed in a fluid, its vibration
energy radiates into the fluid domain. As a result, there is an observable increase
in the kinetic energy of the structure due to the fluid loading. Because of this
kinetic energy increase, the natural frequencies of structures which are subjected
to fluid-loading decrease significantly compared to the natural frequencies of
structures in vacuo. Therefore, through understanding of the interaction between
the elastic plate structures and the fluid loading has been essential to the effective
design of complex structures like ships and submarine hulls. Lindholm et al. [1]
used a chordwise hydrodynamic strip theory approach to study the added mass
factor for cantilever rectangular plates vibrating in still water. Fu et al [2] studied
the dry and wet dynamic characteristics of vertical and horizontal cantilever
square plates immersed in fluid using linear hydroelasticity theory. Ettouney et
al [3] studied the dynamics of submerged structures using expansion vectors,
called wet modes which are finite series of complex eigenvectors of the fluid-
structure system. Recently Kwak [4] presented an approximate formula to
estimate the natural frequencies in water from the natural frequencies in vacuo.
When the structure and the fluid domains become rather complex,
solutions of fluid-structure coupled system can be obtained by finite element
1257
methods. Marcus [5], Chowdhury [6], Muthuveerappan et al. [7] and Rao et al
[8] have successfully implemented the finite element method to predict the
dynamic characteristics of elastic plates in water. Everstine [9] used both finite
and boundary element methods to calculate the added mass matrices of fiilly-
coupled fluid-structure systems.
The above investigations formed the bases necessary to devising passive
and active means for controlling the vibration of as well as the sound radiation
from fluid-loaded plates. Passive Constrained Layer Damping (PCLD) treatments
have been used extensively and have proven to be effective in suppressing
structural vibration as reported, for example, by Jones and Salerno [10], Sandman
[11] and Dubbelday [12]. Recently, Gu and Fuller [13] used feed-forward control
algorithm which relied in its operation on point forces to actively control the
sound radiation from a simply-supported rectangular fluid-loaded plate.
In the present study, the new class of Active Constrained Layer Damping
(ACLD) treatment is utilized as a viable alternative to the conventional PCLD
treatment and Active Constrained (AC) with PCLD treatment (AC/PCLD). The
ACLD treatment proposed combines the attractive attributes of both active and
passive damping in order to provide high energy dissipation-to-weight
characteristics as compared to the PCLD treatment. Such surface treatment has
been successfully employed to control the vibration of various structural members
as reported, for example, by Shen [14] and Baz and Ro [15]. In this paper, the use
of the ACLD is extended to the control of sound radiation from fluid-loaded
plates. Finite element modeling of the dynamics and sound radiation of fluid-
loaded plates is developed and validated experimentally. Particular focus is
placed on demonstrating the effectiveness of the ACLD treatment in suppressing
the structural vibration and attenuating the sound radiation as compared to
conventional PCLD and AC/PCLD.
This paper is organized in five sections. In Section 1, a brief introduction
is given. In Section 2., the concepts of the PCLD, ACLD and AC/PCLD
treatments are presented. In Section 3, the dynamical and fluid finite element
models are developed to describe the interaction between the plate, ACLD and the
contacting fluid. Experimental validation of the models are given in Section 4.
Comparisons between the theoretical and experimental performance are also
presented in Section 4 for different active and passive damping treatments.
Section 5, summarizes the conclusions of the present study.
2. CONCEPTS OF PCLD, ACLD AND AC/PCLD TREATMENTS
Figures (1-a), (1-b) and (1-c) show schematic drawings of the PCLD,
ACLD and AC/PCLD treatments respectively. In Figure (1-a), the plate is treated
1258
Figure (1) - Schematic drawing of different surface treatments (a) PCLD, (b)
ACLD and (c) AC/PCLD.
with a viscoelastic layer which is bonded directly to the plate. The outer surface
of the viscoelastic layer is constrained by an inactive piezo-electric layer in order
to generate shear strain y, which results in dissipation of the vibrational energy of
the plate. Activating the constraining layer electrically, generates a control force
Fp by virtue of the piezo-electric effect as shown in Figure (l-b) for the ACLD
treatment . Such control action increases the shear strain to yj which in turn
enhances the energy dissipation characteristics of the treatment. Also, a restoring
moment Mp=d2Fp is developed which attempts bring the plate back to its
undeformed position. In the case of AC/PCLD treatment, shown in Figure (l-c),
two piezo- films are used. One film is active and is bonded directly to the plate to
control its vibration by generating active control (AC) force Fp and moment
Mp^djFp. The other film is inactive and used to restrain the motion of the
1259
viscoelastic layer in a manner similar to the PCLD treatment of Figure (1-a). In
this way, the AC action operates separately from the PCLD action. This is unlike
the ACLD configuration where the active and passive control actions operate in
unison. Note that in the ACLD configuration, larger shear strains are obtained
hence larger energy dissipation is achieved. Furthermore, larger restoring
moments are generated in the ACLD treatments as compared to the AC/PCLD
treatments as the moment arm d2 in the former case is larger than the moment arm
d3 of the latter case. This results in effective damping of the structural vibrations
and consequently effective attenuation of sound radiation can be obtained.
3. FINITE ELEMENT MODELING
3.1 Overview
A finite element model is presented in this section, to describe the
behavior of fluid-loaded thin plates which are treated with ACLD, PCLD and
AC/PCLD treatments.
3.2 Finite Element Model of Treated Plates
(b) (c)
Figure (2) - Schematic drawing of plate with ACLD/AC/PCLD patches.
Figure (2) shows a schematic drawing of the ACLD and AC/PCLD
treatments of the sandwiched plate which is divided into N finite elements. It is
assumed that the shear strains in the piezo-electric layers and in the base plate are
negligible. The transverse displacement w of all points on any cross section of the
sandwiched plate are considered to be the same. The damping layers are assumed
to be linearly viscoelastic with their constitutive equations described by the
complex shear modulus approach such that G=G (1+T|i). In addition, the bottom
piezo-electric layer (AC) and the base plate are considered to be perfectly bonded
together and so are the viscoelastic layer and the top piezo-electric layer.
The treated plate elements considered are two-dimensional elements
bounded by four nodal points. Each node has seven degrees of fireedom to
describe the longitudinal displacements u, and v, of the constraining layer, U3 and
V3 of the base plate, the transverse displacement w and the slopes w and W y of
the deflection line. The deflection vector {5} can be written as:
{5} = {u„v„U3, V3,W, W ^W y}""
= [{n,} {N3} {Nj {n.} {n,} {N,}^
where {5"} is the nodal deflection vector, {Nj}, {Nj}, {N3}, {N4}, {N5}, {N5} ,,,
and {N5} y are the spatial interpolating vectors corresponding to u„ v„ U3, V3, w,
w^, and Wy respectively. Subscripts ,x and ,y denote spatial derivatives with
respect to x and y.
Consider the following energy functional ITp for the treated plate/fluid
system:
np = I(u-TK+w.-w^+wJdv, (2)
where U is the strain energy, T^ is the kinetic energy, is work done by external
forces, Wp is work done by the back pressure inside the fluid domain, is work
done by the control forces and moments and V is the volume of the plate. These
energies are expressed as follows
I T,dv =X ^ 1 1„ (*' = 5{®T['^p]{®') ’
|w,dV=j{5'f{F},
= 5' Kj 5'
“ V.l^dxdy
„[Ui hi ‘J “ ax^ Ui ‘ hi j “ Sy^J
where i=l for ACLD control or i=3 for AC control (6)
and I WpdV ={5'}’"[n ]{p'}.
where {p®} is the nodal pressure vector of the fluid element. In the above
equation []^], [Mp], {F}, [KJ and [Q] are the plate stiffness matrix, mass matrix,
external forces vector, piezo-electric forces and moments matrix and plate/fluid
coupling matrix as given in the appendix. In equation (6), d3i 32 are the piezo¬
strain constants in directions 1 and 2 due to voltage applied in direction 3. The
voltage is generated by feeding back the derivative of the displacement 5 at
critical nodes such that j where is the derivative feedback gain
matrix and C is the measurement matrix defining the location of sensors.
Minimizing the plate energy fimctional using classical variational methods
such that |anp/a{6®}j = 0 leads to the following finite element equation:
{[K]-»lM.]){5'}-[n]{p'} = {F} (8)
where co is the frequency and [K] = [Kp] + [K^.] is overall stiffriess matrix.
3.3 Finite Element Model of the Fluid
The fluid model uses solid rectangular tri-linear elements to calculate the
sound pressure distribution inside the fluid domain and the associated structural
coupling effects. The fluid domain is divided into fluid elements. Each of
1262
these elements has eight nodes with one degree of jfreedom per node. The
pressure vector is expressed by p = [Nf]{p®} and [NJ is pressure shape function
and {p®} is nodal pressure vector.
Considering the following functional Elf of fluid domain Craggs [16]
where [Kf] and [MJ are the fluid stiffiiess and mass matrices as given in the
appendix. Minimization of equation (9) such that {OTf/^lp^}} = 0 yields the
fluid dynamics as coupled with the structural vibration:
([K,]-o)^[M,]){p'}-<a^[nf{5'} = {0} (10)
The boundary conditions involved are of the form
ap/an = 0, at a rigid boundary
a p / a n = -pp 9^5 / 9 ,
and p = 0.
at a vibrating boundary
at a free surface
where pf is fluid density.
3.4. Solutions of the Coupled Plate/Fluid System
Combining equations (8) and (10) gives
■[K]-co^[m,] -[fi] Ip'l.pl (11)
[Kr]-CD^[M,]J Lp'J k
At low frequencies, the fluid pressure is in phase with the structural
acceleration, i.e. the fluid appears to the structure like an added mass. However,
as the frequency increases the added mass effect diminishes and the damping
effect, i.e. the pressure proportional to velocity, increases. For an incompressible
fluid, the speed of sound c approaches oo, thus the mass matrix of the fluid [MJ
vanishes, and equation (11) can be simplified to
■[K]-cd=[m,] -[fl]] rs'] pi (12)
_ [KfiJkJ k
If the fluid-structure coupled system has free boundary surface, then [KJ is
non-singular [Everstin, 1991] and the nodal pressure vector {p®} can be eliminated
from equation (12) as follows:
{p'}=-o.^[K,nnr{5'} (13)
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Hence, equation (12) yields
([K]-ffl^(K]+[Mj)){8') = {F) (14)
where [MJ is added virtual mass matrix defined by [Muthuveerappan, 1979]
[Mj = [a][K,]-'[£2f (15)
Equation (14) only involves the unknown nodal deflection vector {5®} of the
structure. When {F}=0, equation (14) becomes an eigenvalue problem, the
solution of which yields the eigenvalues and eigenvectors. The nodal pressures
can then be obtained from equation (13) when the nodal displacements are
determined for any particular loading on plate.
4. PERFORMANCE OF PARTIALLY TREATED PLATES WITH
FLUID LOADING
In this section, comparisons are presented between the numerical
predictions and experimental results of the natural frequencies and damping ratios
of a fluid-loaded plate treated with ACLD, PCLD and AC/PCLD. The effect of
the Active Control, Passive Constrained Layer Damping and Active Constrained
Layer Damping on the resonant frequency, damping ratio, attenuation of vibration
amplitude and sound radiation are investigated experimentally. The vibration and
sound radiation attenuation characteristics of the fluid-structure coupled system
are determined when the plate is excited acoustically with broadband frequency
excitation while the piezo-electric layers are controlled with various control gains.
The experimental results are compared with the theoretical predictions.
4.1 Experimental Set-up
Figures (3-a) and (3-b) show a schematic drawing and finite element mesh
of the experimental set-up along with the boundary conditions used to describe
the fluid-structure system. The finite element mesh includes: 24 plate-elements
and 560 fluid-elements. The coupled system has a total of 815 active degree of
freedoms. The aluminum base plate is 0.3m long, 0.2m wide and 0.4inm thick
mounted with all its edges in a clamped arrangement in a large aluminum base.
The aluminum base with mounting frame sits on top of a water tank. One side of
the base plate is partially treated with the ACLD/AC/PCLD and the other side is
in contact with water. The material properties and thickness of piezo-electric
material and the viscoelastic layer listed in Table (1). The size of the combined
piezo-electric and viscoelastic patch occupied one-third of the surface area of the
base plate and it is placed in the middle of plate as shown in Figure (3-b). A laser
1264
sensor is used to measure the vibration of the treated plate at node 27 as shown in
Figure (3-b). The sensor signal is sent to a spectrum analyzer to determine the
frequency content and the amplitude of vibration. The signal is also sent via
analog power amplifiers to the piezo-electric layers to actively control the sound
radiation and structural vibration. The radiated sound pressure level into the tank
is monitored by a hydrophone located at 5.0 cm below the plate center. This
position is chosen to measure the plate mode (1, 1) which dominates the sound
radiation. The hydrophone signal is sent also to the spectrum analyzer to
determine its frequency content and the associated sound pressure levels.
Figure (3) - The experimental set-up, (a) schematic drawing, (b) finite element
meshes.
Table (1) - Physical and geometrical properties of the ACLD treatment
Layer
Thickness(m)
Density (Kg/m^)
Modulus(MPa)
viscoelastic
S.OSxlO’"
1104
30**
piezoelectric
28xl0-‘
1780
* Young’s modulus ** Shear modulus
4.2 Experimental Results
Experimental validation of the dynamic finite element model of the
ACLD/plate system in air has been presented by Baz and Ro [15] in detail. Close
agreement was obtained between the theoretical predictions and the experimental
1265
measurements. The d)mamic finite element model is therefore valid to provide
accurate predictions.
For the uncontrolled treated plate/fluid system, considered in this study,
the experimental results indicate that coupling the plate with the fluid loading
results in decreasing the first mode of vibration fi*om 59.475Hz to 10.52Hz. The
coupled finite element model predicts the first mode of vibration to decrease firom
57.91Hz to 10.24Hz. The results obtained indicate close agreement between the
theory and experiments.
Figure (4-a) shows a plot of the normalized experimental vibration
amplitudes for the fluid-loaded plate with the ACLD treatment using different
derivative feedback control gains. According to Figure (4-a), the experimental
results obtained by using the ACLD treatment indicate that amplitude attenuations
of 1 1.36%, 48.25% and 75.69% are obtained, for control gains of 2500, 5000, and
13500, respectively. The reported attenuations are normalized with respect to the
amplitude of vibration of uncontrolled plate, i.e. the plate with PCLD treatment.
Figures (4-b) display the vibration amplitudes of the plate/fluid system with
AC/PCLD treatment at different derivative feedback control gains. The
corresponding experimental attenuations of the vibration amplitude obtained are
4.6%, 20.29%, 54.04% respectively.
Figure (4) - Effect of control gain on normalized amplitude of vibration of the
treated plate, (a) ACLD control and (b) AC/PCLD control.
Figures (5-a) and (5-b) show the associated normalized experimental
sound pressure levels (SPL) using ACLD and AC/PCLD controllers, respectively.
The normalized experimental SPL attenuations obtained using the ACLD
controller are 26.29%, 50.8% and 76.13% compared to 10.02%, 24.52% and
53.49% with the AC/PCLD controller for the considered control gains. Table (2)
1266
lists the maximum control voltages for the ACLD and AC/PCLD controllers for
the different control gains.
It is clear that increasing the control gain has resulted in improving the
attenuations of the plate vibration and the sound radiation into the fluid domain.
It is evident that the ACLD treatment has produced significant vibration and
sound pressure level attenuation as compared to the attenuations developed by the
AC/PCLD or PCLD treatments. It is also worth emphasizing that the ACLD
treatment requires less control energy than the conventional AC/PCLD treatments
to control the sound radiation from the plate.
Figure (5) - Effect of control gain on normalized sound pressure level radiated
from the treated plate, (a) ACLD control and (b) AC/PCLD control.
Table (2) - Maximum control voltage for the ACLD/ AC/plate system
K.
2500
5000
13500
ACLD
0
21.75 V
31.20V
39.60V
AC
50.40V
76.38V
Figure (6) shows the mode shapes of the first four modes of the treated
plate with and without fluid-loading as obtained experimentally using
STARMODAL package. Figure (7) shows the corresponding theoretical
predictions of the first four mode shapes. Close agreement is found between
experimental measurement and theoretical predictions.
Figure (8) presents comparisons between the theoretical and experimental
natural frequencies and the loss factor of a plate treated with the ACLD and
AC/PCLD for different control gains. Close agreement between theory and
1267
experiment is evident. Note also that increasing the control gain has resulted in
increasing the damping ratio for both ACLD and AC/PCLD treatments. The
comparisons emphasize the effectiveness of the ACLD treatment in acquiring the
large damping ratio to attenuate the structural vibration and sound radiation.
Figure (6) - Experimental results of first four mode shapes of treated plate (a)
without fluid loading and (b) with fluid loading.
Figure (7) - Theoretical predictions of first four mode shapes of treated plate (a)
without fluid loading and (b) with fluid loading.
5. SUMMARY
This paper has presented theoretical and experimental comparisons
between the damping characteristics of plates treated with ACLD and
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conventional AC with PCLD treatments. The dynamic characteristics of the
treated plates when subjected to fluid loading is determined for different
derivative control gains. The fundamental issues governing the performance of
this class of smart structures have been introduced and modeled using finite
element method. The accuracy of the developed finite element model has been
validated experimentally. The effectiveness of the ACLD treatment in attenuating
structural vibration of the plates as well as the sound radiated from these plates
into fluid domain has also been clearly demonstrated. The results obtained
indicate that the ACLD treatments have produced significant attenuation of the
structural vibration and sound radiation when compared to PCLD and to AC with
PCLD. Such favorable characteristics are achieved with control voltages that are
much lower than those used with conventional AC systems. The developed
theoretical and experimental techniques present invaluable tools for designing and
predicting the performance of the plates with different damping treatments and
coupled with fluid loading that can be used in many engineering applications.
♦ PCLD ■ ACLD, K:d=l 3500 □ AC, K:d= 13500
• ACLD,Kd=2500 O AC, Kd=2500
A ACLD,Kd=5000 A AC, Kd=5000
Theoretical Natural Frequency (Hz) Theoretical Damping Ratio
Figure (8) - Comparison between theoretical predictions and experimental results,
(a) natural frequency, (b) damping ratio.
ACKNOWLEDGMENTS
This work is funded by The U.S. Army Research Office (Grant number
DAAH-04-93-G-0202). Special thanks are due to Dr. Gary Anderson, the
technical monitor, for his invaluable technical inputs.
1269
REFERENCES
1. Lindholm U. S., Kana, D. D., Chu, W. H. and Abramson, H. N., Elastic
vibration characteristics of cantilever plates in water. Journal of Ship
Research, 1965, 9, 11-22.
2. Fu, Y. and Price, W. G., Interactions between a partially or totally immersed
vibrating cantilever plate and the surrounding fluid. Journal of Sound and
Vibration, 1987, 118(3), 495-513.
3. Ettouney, M. M., Daddazio, R. P. and Dimaggio, F. L., Wet modes of
submerged structures - part litheory. Trans, of ASMS, Journal of Vibration
and Acoustics, 1992, 114(4), 433-439.
4. Kwak, M. K., Hydroelastic vibration of rectangular plates. Trans, of ASME
Journal of Applied mechanics, 1996, 63(1), 110-115.
5. Marcus, M. S., A finite-element method applied to the vibration of
submerged plates. Journal of Ship Research, 1978, 22, 94-99.
6. Chowdury, P. C., Fluid finite elements for added mass calculations.
International Ship Building Progress, 1972, 19, 302-309.
7. Muthuveerappan G., Ganesan, N, and Veluswami, M. A., A note on vibration
of a cantilever plate immersed in water. Journal of Sound and Vibration,
1979, 63(3), 385-391.
8. Rao, S. N. and Ganesan, N., Vibration of plates immersed in hot fluids.
Computers and structures, 1985, 21(4), 111-1%! .
9. Everstine G. C., Prediction of low frequency vibrational frequencies of
submerged structures. Trans, of ASME, Journal of Vibration and Acoustics,
1991, 113(2), 187-191.
10. Jones, I. W. and Salerno, V. L., The vibration of an internally damped
sandwich plate radiating into a fluid medium. Trans, of ASME, Journal of
Engineering for Industry, 1965, 379-384.
11. Sandman B. E., Motion of a three-layered elastic-viscoelastic plate under
fluid loading. J. of Acoustical Society of America, 1975, 57(5), 1097-1107.
12. Dubbelday, P. S., Constrained-layer damping analysis for flexural waves in
infinite fluid-loaded plates. Journal of Acoustical Society of America, 1991,
(3), 1475-1487.
13. Gu, Y. and Fuller, C. R., Active control of sound radiation from a fluid-
loaded rectangular uniform plate. Journal of Acoustical Society of America,
1993, 93(1), 337-345.
14. Shen, I. Y., Bending vibration control of composite plate structures through
intelligent constrained layer treatments. Proc. of Smart Structures and
Materials Conference on Passive Damping ed. C. Johnson, 1994, Vol. 2193,
115-122, Orlando, FL.
1270
15. Baz, A. and Ro, J., Vibration control of plates with active constrained layer
damping. Journal of Smart Materials and Structures, 1996, 5, 272-280.
16. Craggs, A., The transient response of a coupled plate-acoustic system using
plate and acoustic finite elements. Journal of Sound and Vibration, 1971,
15(4), 509-528.
APPENDIX
1. Stiffness Matrix of the Treated Plate Element
The stiffoess matrix [Kp]; of the ith element of the plate/ACLD system is
given by Baz and Ro [15]:
(A-1)
where [K,l and [KJi denote the in-plane, shear and bending stifj&iesses of
the ith element. These stiffoess matrices are given by:
j = layer 1,2, and 3 (A-2)
and [K.,1 = I jjB,]"[Dj,][B,]dxdy j = layer 1, 2 and3 (A-4)
with G2 denoting the shear modulus of the viscoelastic layer and the matrices [BJ,
b1 = :^
\({N2}-{N4)/d + {N,}
(n,1 +fNj
’ [Bb] =
2{N5},„_
{Naj
[B.] =
{n.
[®^p] =
(N.},. + {N3},.+h{Ns},„
{n.},, + + {N2}^ +{n.) „ +h{N4^,
1271
- 1
0
_ i
1 -
0
Ei
1 n
and fj, 1 EA
1 n
1-vJ
Vj 1 u
Vj 1 u
I_
' « '“Vi
0
0
i'
_o 0 V.
. j=l,2and3 (A-5)
where h = (hi-h3)/2 and d = (h2+hi/2+D) with D denoting the distance from the
mid-piane of the plate to the interface with the viscoelastic layer. Also, Ij
represent the area moment of inertia of the jth layer.
2. Mass Matrix of the Treated Plate Element
The mass matrix [Mp]; of the ith element of the plate/ACLD system is
given by:
(A-6)
where [Mjp]j and [M^Jj denote the mass matrices due to extension and bending of
the ith element. These matrices are given by
[Mi,].=p,h,££({N,}^{N,}+{N3}''{N,})dxdy + p,h,[ [({N3)'"{N3} + {NX{N4))dxdy
+ ip2h2 £ { +{N,}^{N,})dxdy
and [ ], = (p,h, + P3h3 + P3h3) £ £ [ N3 f [ N3 ]dxdy (A-7)
where {NJ = {N,}+{N3}+h{Ns},3 and {N,} = {NJ+{N4}+h{N5},3
3. Control Forces and Moments Generated by the Piezo-electric Layer
3.1 The in-plane piezo-electric forces
The work done by the in-plane piezo-electric forces {Fp}i of the ith
element is given by:
i{5'}-{Fp}rhi££%d>'dy (A-8)
where j=l for ACLD control or j=3 for AC control. Also, Ojp and Sjp are the in¬
plane stresses and strains induced in the piezo-electric layers. Equation (A-8)
reduces to:
1272
='"•1 IKFK]
for k=l, 4
3.2 The piezo-electxic moments
The work done by the piezo-electric moments {Mp}i due to the bending of
the piezo-electric layer of the ith element is given by:
5' .MM
= hj U,e,Ady
(A-10)
Where Gjb and are the bending stresses and strains induced in the piezo-electric
layers. Equation (A-10) reduces to:
Mpxk
x.'
Mpyk
d32
Mp,yk
_ 0
fork=l,..,4
(A-11)
4. Stiffness and Mass Matrices of the Fluid Element
The stiffness matrix [KJj and mass matrix [MJ^ of the ith element of the
fluid system are given by:
[B,rNdv
(A-12)
(A-13)
[N,]>,]dv (A-13)
where [b^ ] = [[N^],, [N^] ^ [N^] and c is the sound speed.
5. Coupling matrix of the Treated Plate/Fluid System
The coupling matrix [Q] of the interface element of the structure/fluid
system can be presented by:
[q]= f [ {N5}''[Nf]dxdy (A-14)
Ja Jb
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1274
ANALYTICAL METHODS II
DYNAMIC RESPONSE OF SINGLE-LINK FLEXIBLE
MANIPULATORS
E. Manoach^ G. de Paz^ K. Kostadinov^ and F. Montoya^
^ Bulgarian Academy of Sciences, Institute of Mechanics
Acad. G. Bonchev St. Bl. 4; 1113 Sofia, Bulgaria
^ Universidad de Valladolid, E.T.S.I.L Dpto. IMEIM.
C/Paseo del Cauce, s/n 47011- Valladolid, Spain
1. INTRODUCTION
The flexible-link manipulators have many advantages over the traditional stiff
ones. The requirements for light-weight and energy efficient robotic arms
could be naturally satisfied by using flexible manipulators. On the other hand
the application of the robotic arm in such activities as positioning in electronic
microscopes and disc-drivers, hammering a nail into a board or playing tennis,
also forces the modeling and control of the dynamic behavior of flexible link
manipulators.
In most cases the elastic vibrations which arise during the motion must be
avoided when positioning the end point of a robotic arm. These are a part of
the reasons that cause a great increase of the publications in this topic in recent
years.
In most papers the flexible robotic arms are modeled as thin linear elastic
beams. In [1-3] (and many others) the Bemouli-Euiler beam theory, combined
with finite-element technique for discretization with respect to the space
variables is used for modeling and control of single-link flexible manipulators.
The same beam theory, combined with mode superposition technique is used
in [4]. Geometrically nonlinear beam theories are used in [5,6] for the
modeling of a single-link and multi-link flexible robotic manipulators,
correspondingly.
Taking into account the fact that robotic arms are usually not very thin and that
the transverse shear could play an important role for dynamically loaded
structures [7] the application of the Bernouli-Euiler beam theory could lead to
a discrepancy between the robotic arm behavior and that one described by the
model.
1275
The aim of this work is to model the dynamic behavior of a single link flexible
robotic arm employing the Timoshenko beam theory, which considers the
transverse shear and rotary inertia. The arm is subjected to a dynamic loading.
As in [3], the viscous friction is included into the model and slip-stick
boundary conditions of the rotating hub are introduced. Besides that, the
possibility of the rise of a contact interaction between the robotic arm and the
stop (limiting support) is included into the model. The beam stress state is
checked for plastic yielding during the whole process of deformation and the
plastic strains (if they arise) are taken into account in the model. The
numerical results are provided in order to clarify the influence of the different
parameters of the model on the response of the robotic arm.
2. BASIC EQUATIONS
2.1. Formulation of the problem
The robotic arm - flexible beam is attached to a rotor that has friction and
inertia. The beam is considered to be clamped to a rotating hub and its motion
consists of two components: “rigid-body” component and a component
describing the elastic deflection of the beam (see Figure 1). The motion of the
flexible beam is accomplished in the horizontal plane and gravity is assumed
to be negligible.
Figure 1. Model of one-link flexible manipulator.
1276
Tip of the beam (with attached tip-mass) is subjected to an impulse loading.
Stick-slip boundary conditions due to Coulomb friction of the hub are
introduced when describing the motion of the beam. In other words, if the
bending moment, about the hub axis, due to the impact is lower than the static
friction torque threshold then the hub is considered clamped and the beam
elastic motion is considered only. When the bending moment exceeds friction
torque threshold this boundary condition is removed, allowing rotation of the
hub and the arm. When the hub speed and kinetic energy of the beam become
again beneath the torque threshold, the hub clamps again.
The possibility of the rise of a contact interaction between the robotic and the
stop is envisaged. In this case, if the hub angle exceeds the limit value, the
robotic arm clamps and a part of the beam goes in a contact with the stop,
which is modeled as an elastic foundation of a Vinkler type.
In view of the fact that the impact loading and contact interactions are
included in the investigations, it is expedient to be considered the rise of
plastic strains in the beam.
2.2. Deriving the equations of motion.
The total kinetic and potential energy of the rotating hub with the attached
beam (described by the Timoshenko beam theory) and a lumped mass at it’s
tip can be expressed as follow:
Ek =^|pA[u(x,t)] dx + |j„[e(t)]^+tMT[u(l,t)f jEl[(()(x,t)] dx
EIl^l +kGA’
dx.
V dx
dx
(1 a,b)
In these equation u(x,t) is the total displacement
u(x,t) = w(x,t)+x0(t) (2)
and w is the transverse displacement of the beam, (p is the angular rotation,
9 is the hub angle, E is the Young modulus, G is the shear modulus. A: is a
shear correction factor, p is the material density, Jh is the inertia moment of
the hub, A=b*h is the beam cross-section area, h is the thickness, b is the
width, I=bh^/12 , 1 is the length of the beam, Mt is the tip mass, t is the time.
1277
Denoting the work of external forces (applied actuating torque T(t) and the
beam loading p(x,t))by
W=T(t)e(t)+ \p{x,t)w{,x,t)dx
0
the Hamilton's principle can be applied:
j5(E,-Ep)dt+ j5Wdt = 0
(3)
(4)
Substituting eqns (l)-(3) into eqn (4) after integrating and including damping
of the beam material, the viscous friction of the hub and the reaction force of
the elastic foundation the following equations of motion can be obtained:
. r . X. i2x de , ,5^w(l,t) ,
(J + Jh + M-j-1 ^ j ^2 ^ ^ ^ ^2
i
JpAx
a^w(x,t)
dt^
dx = T(t)
EI^-^ + kGA
dx^
dw
dx
dw
.d^cp
- 9 “ <^2 - — 2 ~ 0<x</ , t>0 (5a-c)
dt
dt
kGA
+ x-
a^0
'St"
-R(x,t) = p(x,t)
The boundary conditions are:
w(0,t)=(p(0,t)=0
dcpgt) Q
dx
kGA
dw(l,t)
dx
d^w(lt)
dt^
(6a-c)
and the initial conditions are:
wfx, 0) =w^(x), w(x,0) = w° (x) , p(x, 0)=(f^(x), (p(x,0) = 9° (x)
e(t) = e(t) = o, t<t,,p
or 9(0 =
when |9(t)| and kinetic energy of the beam falls simultaneously under the
Coulomb friction thresholds.
In eqns (5) the viscous friction coefficient of the hub is denoted by di, d2 and
d^ are damping coefficients of the beam material, J is the moment of inertia of
the beam about the motor axis and R(x,t) is the reaction force of the stop
disposed from Xj to X2 (xj<X2<l) and modeled as an elastic foundation with
Vinkier constant r:
(7a-d)
(7 e,f)
(7e)’
1278
R(x,t) =
r[w(x, t) - (x)] for x j < x < x^ ; |0l > '
0 for 0<x<x,,X2 <x</; iBi <
The time when the bending moment about the hub axis exceeds the friction
torque threshold is denoted by tsUp^ When t>tsiip this condition is removed
(allowing rotation of the hub and the beam) until the moment when the beam
clamps again.
2.3. Elastic-plastic relationships
The beam stress-strain state is usually expressed in terms of generalized
stresses and strains which are function of x coordinate only. As a unique
yield criterion in terms of moments and the transverse shear force does not
exist according to Drucker [8], the beam cross-section is divided into layers
and for each of them the stress state has to be checked for yielding.
The relation between the stress vector S = and the strain vector
s = |-z ^ I ’ generally presented as
S=[D]8,
where in the case of an elastic material [D] = [D^] =
is the elastic
matrix and f(z) is a function describing the distribution of the shear strains
along the thickness.
On the basis of the von Mises yield criterion, the yield surface is expressed
After yielding during infinitesimal increment of the stresses, the changes of
strains are assumed to be divisible into elastic and plastic parts
As = As^ + As^
where
As
1279
By using eqns (10), (11) and the associated flow rule [9], following Yamada
and others [10], the following explicit relationship between the increments of
stresses and strains is obtained
. AS = [d‘'’]A8
where [-0'^] is the elastic-plastic matrix:
HY\d^]
LjaLiVi
r^r
1 J 1 J L j\asj
1 -
C/2
1—
108]
In this equation H is a function of the hardening parameter. For ideal plasticity
H is equal to zero, while for a wholly elastic material H -> co .
3. Use of mode superposition.
3.1. Rearrangement of the equation of motion.
Let the total time interval T on which the dynamic behavior of the structures is
investigated, be divided into sequence of time increments y+J.
In the numerical calculations the following dimensionless variables are used:
x = xl U w = w//, t =tl / c c = / p
and then omitting the bars, and after some algebra, the governing equations
can be written in the following form:
d'e de
- T- + C, -
d t^ ' d t
--cJ w(l,t)- f(p(x,t)dx =C3 T(t)+ fxp(x,t)dx
tv 0 /Vo /
09 0^9 -I c/w I
(15 a-c)
0^W 01V 0^iv 09
dt^ ^ dt 1 0x ^ dx
= -p~G{-G^
where a=12P/h^ , p~kG/E, p=pI/(EA), c,= d^l/(cJ}{), Arp/(JHp) C3=
P/(c^J}^ ), C4~d49/EI, C3=djl/EA, =5/6. The nonlinear force due to the
reaction of the foundation is denoted by G^ = R(x,t).l/EA and Gf and G2 are
the components of the so-called non-linear force vector Gp( .G^} which is
due to the inelastic strains. It has the presentation (see [7] ):
1280
3.2 Mode superposition method
The l.h.s. of eqns (15 b,c) is a linear form and therefore the mode
superposition method can be used for its solution. As the eigen frequencies
and the normal modes of vibrations of an elastic beam do not correspond to the
real nonlinear system, these modes are called "pseudo-normal" modes.
Thus, the generalized displacements vector v = |a”^ (p, w| is expanded as a
sum of the product of the vectors of pseudo-normal modes v^ and the time
dependent functions q„(t) as
(17)
n
The analytically obtained eigen functions of the elastic Timoshenko beam are
chosen as basis functions (see APPENDIX). When the tip mass is attached to
the beam the eigen functions of the system used in the mode superposition
method are preliminary orthogonalized by standard orthogonalization
procedure as it is mentioned in the APPENDIX.
Substituting eqn (17) into (15 b,c), multiplying by v^(x), integrating the
product over the beam length, invoking the orthogonality condition (see (A9) )
and assuming a proportional damping for the beam material
|(c4(pf, +c^wl)rdr the following system of ordinary differential
equations (ODE) for 0(t) and qn(t) is obtained:
^(f) + c, (9(0 = Cj 2] [«'» (1) - ]?»
^ »
+ C3(r(0 + P(0)
(18a, b)
9„(o+2f„ffl„9„(o+®,k,(o=-‘y„ -'f„&(o+&!’
In this equations
1
11 V f
Y„ = J(p„ 6„ = = jxw,Xx)ck, P(t)= \xp{x,t) dx ,
1 1
g;(t)= |G'’(x,t)v„(x)dx, g',(t)= jG'(x.t)w„(x)dx, co„ are the eigen
0 0
frequencies of the elastic clamped beam and are the modal damping
parameters.
The initial conditions defined by eqns (7 a-d) are transformed also in terms of
q„(0), and ?„(0)
9„(0) = 9„“, 4„(0) = 4°,
q°„ = J(w°w„ t-a'VVjdx, (j“„ = j(w"w„ +a-'(p°(p„)dx, (19a-d)
0 0
The obtained system of nonlinear ordinary differential equations is a stiff one
and it is solved numerically by the backward differential formula method, also
called the Gear's method [11].
The rise of plastic strains is taking into account by using an iterative procedure
based on the "initial stresses" numerical approach [7].
4. RESULTS AND DISCUSSION
Numerical results were performed for the robotic-arm with the same material
and geometrical characteristic as these given in [3] in order to make some
comparisons. Model parameters are: E=6.5xl0 Pa ,v=0.2, Cp=2.6xl0 Pa,
1=0.7652, b=0.00642 m, h=0.016 m, p=2590 kgW , Mt=0,153 kg, Jh=0.285
kgm^ (Jh is not defined in [3].)
The aim of the computations is to show and clarify the influence of the elastic
or elastic-plastic deformation on the motion of the robotic-arm, to demonstrate
the effect of the hub friction, slip-stick boundary conditions and the contact
interaction between the beam and the stop.
1282
Only impact loading on the beam is considered in this work, i.e. dynamic load
p(x,t) and applied torque T(t) are equal to zero. The impact loading is
expressed as an initial velocity applied to the tip of the beam 0.95 < x < 1 .
Nine modes are used in expansion (15) but the results obtained with number of
modes greater than nine are practically indistinguishable from these shown
here.
For all calculations the material damping is equal to 8% of the critical
damping.
The results for the rotation of the hub of the flexible manipulator with an
attached mass at its tip -1 and without an attached mass -2 are shown in Figure
2. The hub friction is not considered. The beam is subjected to an impact
loading with initial velocity =-1.95 m/s. As can be expected, the hub angle
increases much faster in the case of the beam with an attached mass. The
corresponding beam deflections are presented in Figure 3. The results obtained
are very close to these obtained in [3] (Fig. 7 and 8 in [3]). The frequencies of
forced vibrations obtained in [3], however, correspond to the beam without an
attached mass.
0, rad
-4.00 I - ^ ^ ^ ^ - 1 - ^ ^ i ^
0,00 l.OO 2.00 3.00 4.00 5.00
t, sec
Figure 2. Hub response without
viscous friction. 1 - beam with an
attached mass; 2- without an attached
mass
w, m
Figure 3. Deflection of the tip of the
beam without viscous friction of the
hub. 1 - beam with an attached mass;
2 - without an attached mass
1283
The influence of the hub friction on the flexible manipulator response can be
seen in Figure 4. The viscous friction is set di=0.1 Nms, the static Coulomb
friction threshold is equal to 0.06 Nm and three cases of the hub slip-stick
threshold are tested: |e| stick = 0-005 rad.s'^ - 1, |0| stick = 0.0085 rad.s'^ ,
stick = 0.01 rad.s'^ For this initial velocity (w° =-1.95m/s) the hub slips
very fast from the initial clamped state and the beam begins to rotate. As can
be expected, the consideration of the viscous friction of the hub leads to a
decrease of the angle of rotation of the beam and changes the linear variation
of 0 with time. The results show also that the value of the hub speed
threshold |0| stick exercises an essential influence on the motion of the
rotating system.
When |e| =0.005 rad.s' the
beam sticks at t = 2.602 s after that
the hub periodically slips and sticks
which also leads to damping of the
motion. When joj stick “ 0.0085
rad.s'^ the start of sticking occurs at
t = 1.7207 s and after t=2.417 s the
hub clamps with short interruptions
till t=3.4 s when due to the elastic
vibrations it snaps in the direction
opposite to w°» clamps again at
3.679s, slip at 4.5s, and finally
clamps at t=4.5s. When joj stick
=0.0085 rad.s"' the sticking begins
at t= 1.0525s and very fast (at t«2 s)
the beam clamps with 0=-O.587 rad.
In order to observe the occurrence of the plastic deformation the beam was
subjected to impulse loading having larger values of initial velocities. In
addition, the contact interaction between the beam and the stop was
considered.
The beam-tip deflection in the presence of a contact with the stop disposed at
X €[0.16, 0.263] and initial velocity w° =-15.95 m/s is shown in Fig. 5.
0,rad
di=0.1Nms. 1- |0| stick”0-005rad.s"',
2 - |0| stick = 0-0085 rad s"' ;
3- |0| stick =0-0 is-'
1284
In order to reduce the computational time the limit value of 0 was chosen
e^j^p=0.0025 rad. When this value was reached the problem was automatically
resolved with new initial conditions 0=0, w°=w(Xstop,tstop), etc.
As can be seen, the presence of the contact interaction during the process of
motion of the beam due to the elastic support for x e[0.16, 0.263] leads to a
decrease of the amplitudes of vibrations in the direction of the stop in
comparison with the amplitudes in the opposite direction. The variation of the
beam displacements along the beam length for the first 0.8 s of motion is
shown in Fig. 6. It must be noted that in this case of loading a plastic yielding
occurs. It is assumed that beam material is characterized by an isotropic linear
strain hardening and H=0.5. The plastic yielding occurs simultaneously with
the contact interaction at t=0.01366 s at the clamped end of the beam. At
t=0.0186 s the plastic zone spreads to x = 0.158 and at t=0.08767 if covers the
length to X =0.3 1 . The last points that yields are x =0.55, 0.61 at t=l . 1 19 s.
Seven layers along the beam
thickness, symmetrically disposed
about the beam axis was checked for
yielding (N2=7) but the plastic zone
has reached the second and 6th
layers only at the clamped end of the
beam (x =0). In all other point along
beam length the plastic yielding
occurs only at the upper and lower
surface of the beam. The plastic
strains are small and the response of
the beam is not very different from
the wholly elastic response.
Nevertheless, the appearance of such
kind of plastic deformations in the
structures used for the precise
operations must be taken into
account in the manipulator self
calibration procedure.
w,m
0.40
0.30
0.20
0,10
0.00
-O.IO
-0.20
-0.30
0.00
1.00
2.00
3.00
4.00 5.00
t, sec.
Figure 5. Deflection of the tip of the
beam with time in the case of a
contact with the stop. r=5.5xl0^ Pa
5. CONCLUSIONS
In this work a model describing dynamic behavior of a deformable beam
attached to a rotating hub that has friction and inertia is developed. The
Timoshenko beam theory is used to model the elastic deformation of the beam.
1285
Figure 6, Variation of the elastic-plastic beam displacement along the beam
length with time in the case of a contact with the stop.
The slip-stick boundary conditions are also incorporated into the model
The possibility of the rise of undesired plastic deformations in the case of a
high velocity impact on the clamped robotic arm, or in the case of a contact
with limiting support (stop) is included into the model.
The analytically obtained eigen functions of the elastic Timoshenko beam
vibrations are used to transform the partial differential equations into a set of
ODE by using the mode superposition method. This approach minimizes the
number of ODE which have to be solved in comparison with another
numerical discretization techniques (finite elements or finite difference
methods).
The results obtained show the essential influence of elasticity on the robotic-
arm motion.
The model will be used to synthesise a control of one link flexible
manipulators and for a self calibration procedure when plastic deformation
would occur.
Acknowledgments
The authors gratefully acknowledge the financial support from EC Copernicus
Program under the Project ROQUAL CIPA CT 94 0109.
The first author wishes to thank the National Research Fund for the partial
financial support on this study through Contract MM-5 17/95.
1286
REFERENCES
1. Bayo, E. A finite-element approach to control the end-point motion of a
single -link flexible Robert. J. Robotic System, 1987, 4„ 63-75
2. Bayo, E. and Moulin, H., An efficient computation of the inverse dynamics
of flexible manipulators in the time domain. IEEE Proc Int. Conf. on
Robotics and Automations, 1989, 710-15.
3. Chapnik, B.V., Heppler, G.R., and Aplevich, J.D. Modeling impact on a
one-link flexible robotic arm. IEEE Transaction on Robotics and
Automation, 1991,7,479-88.
4. Liu, L. and Hac, A., Optimal control of a single link flexible manipulator.
Advances in Robotics, Mechatronics, and Haptic Interfaces, 1993, DCS-
49, 303-13.
5. Wen, J.T., Repko, M. and Buche, R., Modeling and control of a rotating
flexible beam on a translatable base. Dynamics of Flexible Multibody
Systems: Theory and Experiment, 1992, DCS-37, 39-45.
6. Sharan, A.M. and Karla, P., Dynamic Response of robotic manipulators
using modal analysis. Meek Mach. Theory, 1994, 29, 1233-49.
7. Manoach, E. and Karagiozova, D. Dynamic response of thick elastic-plastic
beams. International Journal of Mechanical Sciences, 1993, 35, 909-19
8. Drucker, D.C. Effect of shear on plastic bending of beams. J. of Applied
Mechanics, 1956, 23, 515-21
9. Hill, R. Mathematical Theory of Plasticity, 1950, Oxford University Press,
London.
10. Yamada, Y., Yoshimura, N. and Sakurai T., Plastic stress-strain matrix and
its application for the solution of elastic-plastic problems by the finite
elements, Int. J. of Mechanical Sciences , 1968, 10, 343-54
11. Gear, C.W., Numerical initial value problem in ordinary differential
equations, 1971, Prentice-Hall, Englewood Cliffs, NJ.
12. Abramovich, H. Elishakoff, 1. Influence of shear deformation and rotary
inertia on vibration frequencies via Love’s equations. J. Sound Vibr., 1990,
137, 516-22.
1287
APPENDIX: NORMAL MODES OF FREE VIBRATIONS OF A
CLAMPED TIMOSHENKO BEAM WITH AN ATTACHED MASS,
Equations (5 b,c) can be decoupled, transforming them into two fourth order
equations [12] as regards cp and w.
Solving this equations (with p=0 and R=0) and using the boundary conditions
(6) (with 0=0) , the equations of the frequencies and forms of vibrations of the
beam are obtained.
Introducing following denotations
1/2 (A1 a-e)
^2„={®,^(l + P) + [®:a + P)^+4(B,^(a-pco,^,)]''^} /2,
/l„ =(4+CoJP)/5,„, /2„ /3„ =(4-®SP)''-S!».
the frequencies of free vibrations are determined as roots of equation:
a) In the case >0 i.e. co < a / p the frequencies equation is:
^11^22 ”^12^21 “ ^ > (-^2)
where
^11 ~ /2n‘^2n ^12 ~ ■^/2«‘^2« ®^^(‘^2«)
h\ = -f\nM.Sxn ) “ /2„ sin(52„ ) + s^„sh(si, ) + 52„ sin(52„ ) + , . .
(A3 a-d)
M‘^lnCh(^lJ + ‘^2« cos(,y2J]
*22 = /2j=OS(^2«)-cll('SlJ] + 4^S,„ch(i,„)-52„ COS(52„) +
J\n
+ sl„ sin(i2„)]
Jin
and the modes of vibrations are:
w„W = -S„
1
' b
^(ch(i,„x) - cos(j2„x)) + sin(i2„x) - ^sh(i|„x)
J\»
L^i
(A4 a,b)
(/,„sh(,s,„x) + sin(,y2«^)) + fi,, icos{s^„x) - ch(s,„x)
b) In the case 5^,, <0 i.e. > a / P the eigen frequency equation (A2) has
the following presentation:
1288
b\\ = + /2„i2„ C0S(J2„)
bn = ?i„/2nSin(?i„) - /2„i2n sin(j2„)
621 = /3„sin(ii„) - /2„ sin(52„) -?i„sin(Ji„) + S2„ sinC^jJ -
>.[J,^„COS( j,„ ) - sl„ C0S(i2„)]
*22 = /2„[cOS(S2„) - COS(J,„)] - ^ S,„COS(?,„) + S2„ COS{s-^„) -
J\n
- sl„ sin(52„)]
7 In
sh = -5?„
and the of vibrations are:
w„(x) =
f b
sin(j2„2c) - ^sin(?,„x) - ^(cosCij,,*) - cos(j,„x))
/i„ *11
'P„W = -S»
(A5 a-e)
(A6a-b)
/j,, (cos(i2„x) - cos(^|„x)) - ^(/3„sin(i,„x) - f^„ sin(jj„x))
0,,
When a mass is not attached at the beam tip the following orthogonality
condition is fulfilled:
1
p fO, n^ m;
(a _
J [1, n - m.
(A7)
and when an attached mass is considered the modes are orthogonalized by
standard orthogonalization procedure.
The constants are obtained from condition (A7).
1289
1290
Wave Reflection and Transmission
in an Axially Strained, Rotating Timoshenko Shaft
B. Kang ‘ and C. A. Tan ^
Department of Mechanical Engineering
Wayne State University
Detroit, Michigan 48202, U. S. A.
Abstract
In this paper, the wave reflection and transmission characteristics of an axially strained,
rotating Timoshenko shaft under general support and boundary conditions, and with geometric
discontinuities are examined. The static axial deformation due to an axial force is also included
in the model. The reflection and transmission matrices for incident waves upon these point
supports and discontinuities are derived. These matrices are combined, with the aid of the
transfer matrix method, to provide a concise and systematic approach for the free vibration
analysis of multi-span rotating shafts with general boundary conditions. Results on the wave
reflection and transmission coefficients are presented for both the Timoshenko and the simple
Euler-Bemoulli models to investigate the effects of the axial strain, shaft rotation speed, shear
and rotary inertia.
‘ Graduate Research Assistant. Tel: +1-313-577-6823, Fax: +1-313-577-8789. E-mail: kang@feedback.eng.wayne.edu
’ Associate professor (Corresponding Author),Tel: +1-313-577-3888, Fax:+1-313.577-8789. E-mail: tan@tan.eng.wayne.edu
Submitted to: Sixth International Conference on Recent Advances in Structural Dynamics, Institute of Sound and Vibration,
Southampton, England, July, 1997
1291
A.
NOMENCLATURE
Area of shaft cross section [m^]
do
Diameter of shaft cross section [m]
C
Generalized coordinate of an incident wave [m]
Cdt {Ct)
Translational damping coefficient [N-sec/m]
Cdr (Cr)
Rotational damping coefficient [N-m-sec/rad]
Co
Bar velocity [m/sec]
Cj
Shear velocity [m/sec]
D
Generalized coordinate of a transmitted wave [m]
E,G
Young’s and shear modulus [NW], respectively
I
Lateral moment of inertia of shaft [m'^]
Jm (Jm)
mass moment of inertia of a rotor mass [kg-m"^]
K
Timoshenko shear coefficient
Knikr)
Rotational spring [N/rad]
Kt (kd
Translational spring [N/m]
t
Length of shaft [m]
M{m)
Mass of rotor [kg]
P
Axial force [N]
rij , ti]
Reflection and transmission coefficients, respectively, i - 1 positive traveling
U{u)
wave; i = 2 negative traveling wave; jr = 1 propagating wave for Cases 11 and
/V; 7 = 2 attenuating wave for Cases II and IV. Both 7 =1 , 2 for propagating
wave for Case I
Transverse displacement [m]
X-Y-Zix-y-z)
Reference frame coordinates [m]
a
{K-G)IE
Rotation parameter, see Eqn. (Id)
£
Pl{E-A), axial strain
e'
Non-dimensional axial load parameter, see Eqn. (13b)
T, f (r, y)
Wavenumber [m‘‘]
nAi)
See Eqns. (20a, b), (22a, b) and (24a, b)
p
Mass density of shaft [kgW]
G
Diameter ratio between two shaft elements
57, (ft))
System natural frequency for Timoshenko model [rad/sec]
a
System natural frequency for Euler-Bernoulli model [rad/sec]
Q.
Rotation speed of shaft [rad/sec]
W (¥)
Bending angle of the shaft cross-section [rad]
subscript L, r
The left and right side of a discontinuity, respectively.
superscript -,+
Negative and positive traveling waves, respectively, when used in C and D.
Note: Symbols in
Otherwise denotes quantities on the left and right side of a discontinuity,
respectively
parenthesis are the corresponding non-dimensional parameters.
1292
1. INTRODUCTION
The vibrations of elastic structures such as strings, beams, and plates can be described in
terms of waves propagating and attenuating in waveguides. Although the subject of wave
motions has been considered much more extensively in the field of acoustics in fluids and solids
than mechanical vibrations of elastic structures, wave analysis techniques have been employed to
reveal important, physical characteristics associated with vibrations of structures. One advantage
of the wave technique is its compact and systematic approach to analyze complex structures such
as trusses, aircraft panels with periodic supports, and beams on multiple supports [1]. Previous
works based on wave propagation techniques have been well documented in several books [2-4],
Recently, Mead [5] applied the phase-closure principle to determine the natural frequencies of
Euler-Bemoulli beam models. A systematic approach including both the propagating and near-
field waves was employed to study the free vibrations of Euler-Bemoulli beams [6].
High speed rotating shafts are commonly employed in precision manufacturing and power
transmission. Despite the usefulness of the wave propagation method in structural vibrations,
applications of this technique to study the dynamics and vibrations of a flexible shaft rotating
about its longitudinal axis have seldom been considered. The purpose of this paper is to examine
the wave reflection and transmission [6] in an axially strained, rotating Timoshenko shaft under
various support and boundary conditions. The effect of the axial load is included by considering
the axial static deformations in the equations of motion. This paper is a sequel to another paper
in which the authors discuss the basic wave motions in the infinitely long shaft model [7].
Although there have been numerous studies on the dynamics and vibration of rotating shafts,
none has examined the effects of axial strains (which cannot be neglected in many applications)
on the vibration characteristics of a Timoshenko shaft under multiple supports. Modal analysis
technique has been applied to study the vibration of a rotating Timoshenko shaft with general
boundary conditions [8, 9], and subject to a moving load [10]. Recently, the distributed transfer
function method was applied to a rotating shaft system with multiple, geometric discontinuities
[11], The wave propagation in a rotating Timoshenko shaft was considered in Ref. [12]. Other
major works on the dynamics of rotating shafts have been well documented in Refs. [13-15].
This manuscript is organized as follows. Governing equations of motion [16] and basic wave
solutions for the Timoshenko shaft are outlined in Section 2. Each wave solution consists of four
wave components: positive and negative, propagating and attenuating waves. In Section 3, the
wave reflection and transmission matrices are derived for the shaft under various point supports
and boundary conditions. The supports may include translational and rotational springs and
dampers, and rotor mass. Results are presented for both the Timoshenko and the simple Euler-
Bemoulli models to assess the effects of axial strain, shaft rotation, shear and rotary inertia. The
wave propagation across a shaft with geometric discontinuities such as a change in the cross-
section is examined in Section 4, and the wave reflection at a boundary with arbitrary support
conditions is considered in Section 5.
With the wave reflection and transmission matrices as the main analytical tools, it is shown in
Section 6 how to apply the current results together with the transfer matrix method to analyze the
1293
free vibration of a rotating, multi-span Timoshenko shaft system in a systematic manner. The
proposed approach is then demonstrated by considering the free vibration of a two-span beam
with an intermediate support.
2. FOKMULATION AND WAVE SOLUTIONS
Consider a rotating shaft subjected to axial loads and with multiple intermediate supports and
arbitrary boundary conditions, as shown in Fig. 1. Including the effects of rotary inertia, shear
deformations, and axial deformations due to the axial loads, the uncoupled equations of motion
governing the transverse displacement u and the slope \}/ due to bending can be derived in the
following non-dimensional form
.. . d'^u „ d^u
£
+16a(l + £)(l-i-e - )^r^ = 0 ,
a dr
U lA £ . d~U
■2ri3^ + a— -16£{1 + £--)^^
a dz'^
-l-16a(l-l-£)(l + £-— = 0 ,
a dr
u = — z = — r = — T = ^ ■
a.' a.' T. ^ \ KG
Figure 1. A rotating Timoshenko shaft model subject to axial loads and with general boundary conditions.
1294
(Id)
a
KG
E ’
E
Note that u and y/are the measurements in the complex plane, that is u=ux+iuy and \i/=\}fx+iYr
E denotes the Young’s modulus, p the mass density, As the area of the cross section, ao the
diameter of shaft, K the Timoshenko shear coefficient, G the shear modulus and Q the constant
angular velocity of the shaft. Details of deriving these equations of motion are found in Ref.
[16].
Assuming and substituting the following wave solutions into Eqns. (la) and (lb)
= (2a)
= (2b)
and defining the non-dimensionalized wavenumber 7 and system natural frequency w gives the
frequency equation, Eqn. (3a); see Ref. [7],
y = r^o .
_ CO a,
KG .
(j) = - a. = - is known as the shear velocity) .
where,
7“* -A7" 4-5 = 0 ,
A = (1 -i- a)a) ^ - 2j3 cu “ 16£ (1 + e - ) ,
5 = a ot)^ - 2j3 u) - 16a (1 4- £)(1 4- e - •^)
The four roots of Eqn. (3a) are
7 = ±-^[a ± -x/a^ -45 .
(2c)
(2d)
(3a)
(3b)
(3c)
(4)
In general, 7 is complex. Let (o be real. It can be shown that, with a > 0 and £ the axial strain
of the elastic solid, the discriminant A^ -45 is positive semi-definite for most engineering
applications. Hence, it is possible to classify the wave solutions into four distinct cases. Note
that one may study the wave propagation by considering only a single general form of the wave
solution. However, the classification procedure identifies the coupled modes of vibration of the
1295
Timoshenko shaft model and provides a better understanding on how each wave solution governs
the wave motions [7]. Based on the algebraic relationships between A and B, the four valid wave
solutions are obtained as follows.
Case / ( A > 0 and B > 0 );
«(z, t) = + C- (5a)
V(z, t) = (CJ.e-'*’" + )e® (5b)
Case !I{A>0 andB<0):
«(z, t) = (Ce-"''' + + C:^e-^'-‘ + (6a)
yz(z,t) = (c;,^-'^" - c;,/" + + 0;^/=- )£® (6b)
CaseIIIiA<0mdB>0):
uiz,t) = (C>-''' + + C:,«")«'“ (7a)
Vr(z,0 = (C;,a-" + C;,ef>' + + C;je'=')«® (7b)
Coje/V(A<0 andB<0):
k(zA) = (C,>-^'^ +C,>’‘'- + C>-'''=‘ + C,V'’'*')«'® (8a)
V/(z,«) = (C;,e-f- + (8b)
where,
f, =^(|A| + V-5’-4|B|f, r. (9^. W
r, = + 4|B| + |A|)^ , r, + 4|B| - |A|j' , (9c,d)
and the coefficients C* and C" denote positive- and negative-travelling waves from the origin of
disturbance, respectively. Important remarks on the basic wave propagation characteristics are
summarized from [7]. First, the wave solution of Case III does not exist in the real frequency
space since this type of solution represents a situation in which none of the wave components can
propagate along the waveguide. Therefore the study of Case III is excluded in the present paper.
Second, the vibrating motion of the shaft model in Case I is predominately pure shear [17] which
1296
is unique for the Timoshenko shaft model, while in Case U and Case IV the flexural mode and
the simple shearing mode, which are corrected by including the rotary inertia and shearing effects
in the formulation, dominate. Third, when the shaft rotates at a very high speed and/or the shaft
is axially strained by tensile loads, the wave solution of Case IV governs the vibrating motion of
the shaft model in the low frequency range.
For comparison, the parameters A and B in the simple Euler-Bernoulli beam model are
(10a)
(10b)
where, the non-dimensionalized natural frequency 6) is defined as
is known as the bar velocity).
(10c)
Note that, because B is negative, wave solutions of Case I and Case III do not exist.
In general the displacement and the rotation of an infinitesimal shaft element consist of four
wave components as shown by Eqns. (5a-8b). Once the displacement and the bending slope are
known, the moment M and shear force V at a cross section can be determined from
M = EI^,
dz
(ll)
(du ^
y = J .
(12)
Moreover, the kinematic relationship between the transverse displacement and the slope due to
bending is
S^u 3^u . ,3 r
3r~ 3z^ 3z '
(I3a)
where e' denotes the effects of the axial force and is defined as
e = l + e — .
a
(I3b)
1297
3. WAVE REFLECTION AND TRANSMISSION AT SUPPORTS
When a wave is incident upon a discontinuity, it is transmitted and reflected at different rates
depending on the properties of the discontinuity. Consider a rotating Timoshenko shaft model
supported at ^ = 0 ; see Fig. 2. The support simulates a bearing modelled by linear, translational
and rotational springs, dampers, and a rotor mass which typically represents a gear transmitting a
torque. Based on Eqns. (5a-8b), group the four wave components into 2x1 vectors of positive¬
travelling waves and negative-travelling waves C" , i. e..
Recall that, depending on the system parameters, the rotating Timoshenko shaft model has four
(practically three) different wave solutions in the entire frequency region as described in Eqns.
(5a-8b). Thus C^ and C, in the above expression do not always correspond to propagating and
attenuating wave components, respectively. When a set of positive-travelling waves is
incident upon the support, it gives rise to a set of reflected waves C~ and transmitted waves .
These waves are related by
C'=rC"
(15)
D" = tC",
(16)
where r and t are the 2x2 reflection and transmission matrices respectively and are expressed as
r =
'^u
^12*
(17)
/2!
''22.
’^11
^12
t =
(18)
/2I
^22 _
From Eqns. (5a-8b), suppressing term and excluding Case III, the displacements u~ and
and the bending slopes y/~and y/'^ at the left and right of z = 0, respectively, can be expressed
in terms of the wave amplitudes of the displacement. For convenience, the over-bar (•) on the
wavenumbers is dropped hereafter.
Case 1 {A >0 andB>0);
«-(z) = C, + c;,£'^==, (19a)
W'U) = -n,C:,e-‘^“ -77, c;, +T7,C>-'>''- (19b)
1298
M.Jm
Figure 2. Wave motion at a general support (the disk may be considered as a gear transmitting a torque).
where,
_ 72-®'
T/i — / » ~ / •
r,£ 72^
// ( A > 0 and B < 0 ):
«-(z) = c;, + C'e"''' + ,
i^-(z) = 7,, C, >-'"■= -77,C,-^r''''= +)7,C*e-''“ -TJ,C,;;e"=S
k*(z) = A>“’'" + AV’‘.
where
=
r,£'
> Ba “
CaselV{A<0a.ndB<0):
(19c)
(19d)
(20a, b)
(21a)
(21b)
(21c)
(21d)
(22a, b)
1299
the following set of matrix equations can be established for each Case.
Case / ( A > 0 and 5 > 0 ):
n n
r 1
1 1
. ri n
c^ +
rC^=:
L^i 12]
-^2j
-im, Ic+r IrC"
KYi-tIi) i(r2“7?2)J [-KYi-Vi) -i(.Y2-Tl2)\
(28a)
ri,(k,-J„,co^) + iTi,{c,0)~r,) 0)") + 1772 72 ) ^28b)
ik^~mO)-) + i{c,co + r^-r]^) (fc, -mtt>") + z’(c,® + 72 -772) J
1300
Case // ( A > 0 and S < 0 ):
1
.^1
1
1
L^i
1
^2
tc^
1
1
n
-ir,T7,
[i(r, -T7i) r2-j772_
C -r
-^r, -77,)
~ (r, - it], )_
(29a)
r},(k^-J„^0)-) + irij (c^co-r^)
(k, -mco^) + i(c,Q) + r,-T],)
Tl2(k,-J„,co^-r2) + iV2CrO}
{kj -m(0^ +r2)+/(c,Ct}- 772)
tc",
(29b)
Case /y(A<0and5<0);
'1 11
■ 1 1 ■
rC" =
' 1 r
Jli Hi.
-^2 -^1.
jii ^1-
■-1X2772 -£t,77, ■
C'*
-1X2772
KT.-ni) r, -i77,_
-r
— i(r2 — 772 )
-(r, - in,).
(30a)
T]2ik^-J„^(0'") + iT]^{c,C0-y2)
{k, -m(D‘) + i(c,aj + r2 -772)
riiik,-J„co^-y,) + iT],c^co'
{k, - mco^ + r, ) + i{c,(0 -rij)
tC",
(30b)
where Eqns. (15) and (16) have been applied in all Cases. Note that in Eqn. (27a), it is assumed
that the rotational spring at the support is attached to the cross section of a shaft element such
that the rotational spring responds only to the slope change due to rotation of the cross section
and not the total slope change of the neutral axis of the shaft model. This assumption allows the
shearing motion of the shaft element at the support. Note also that the effect of axial loads on the
shear force at the support is neglected since the contribution of axial loads to the shear force at
the support or boundary is small compared to the shear force due to the flexural motion of the
shaft element. Exact moment and force balance conditions at boundaries for a rotating
Timoshenko shaft element subjected to axial loads can be found in Ref. [16].
The corresponding matrix equations for the simple Euler-Bernoulli shaft model are shown in
Appendix I. Solving the set of matrix equations simultaneously for r and t gives the elements of
the reflection and transmission matrices for each Case. The general forms of solutions to these
sets of equations for each Case is not presented in this paper due to space limitation. However
one can obtain the solutions in either closed-form or numerically. Note that in Case II and Case
IV, the first columns of r and t are the reflection and transmission coefficients due to incident
propagating wave components, and the second columns are due to an incident attenuating wave
component which is generally termed as near-field since this type of wave decays exponentially
with distance. When the distance between the origin of disturbance and the discontinuity is very
1301
large, these attenuating wave components can be neglected. However, as mentioned by many
authors, for example Graff [2], attenuating waves play an important role in wave motions by
contributing a significant amount of energy to the propagating wave components when a set of
propagating and attenuating waves are incident at a discontinuity and, in particular, when the
distances between the discontinuities are relatively small, as in the case of closely-spaced multi¬
span beams. In this paper, near-field components are included. In what follows, the effects of
the point supports on the reflection and transmission of an incident wave are studied. For
comparison, the results are obtained for both the Timoshenko and the simple Euler-Bemoulli
models, which hereafter, for brevity, are denoted by TM and EB, respectively. The system
parameters used in the numerical results are taken from Ref. [10]; ao - 0.0955 m, p = 7700
kgW, K = 0.9, E = 207x10^ Wnf, G = 77.7x10^ N/ml
3.1. Wave reflection and transmission at rigid supports
Consider two cases: the simple support and the clamped support. The r and t are solved and
shown as follows.
• Simple support (k, = oo, = m= c, = - 7„, =0)
Case 7 ( A > 0 and B > 0 ):
r =
_ 1 _
(72 -7i)(r 1/2+6;')
7i(6>"-72)
72(7? -6;-)
7i(6;’-72)
(31a)
t =
_ 1 _
(72-7i)(7i72+6)-)
72(7?-®')
Case 77 ( A > 0 and S < 0):
_ 1 _
_ 1 _
(iT,-r,xr,r,-!V)
Case 7V(A<0andB<0):
_ 1 _
7i(6)"“72)
7i(72 -6)')
r,(r,H®-) 1
r,{r^ + co^)'
(31b)
(32a)
(32b)
(33a)
1302
The corresponding reflection and transmission matrices for the EB model are listed in App. I
Figures 3 and 4 plot the moduli (magnitudes) of the reflection and transmission coefficients
for the simple and clamped supports. The finite cutoff frequencies, above which all waves
propagate, are also marked in the figures. Thus, for the TM model, the wave motions change
from Case 11 to Case 1 when (0>C0^ {0)^-4- in Fig. 3, co, = 4.24 in Fig. 4; O), is slightly altered
by rotation speed and axial load). The results show that, at low frequencies (cD < 0.1 = 3156
rad/sec), the wave reflection and transmission coefficients of the TM model agree well with those
of the EB model for both support conditions. However, as the frequency increases, the wave
propagation characteristics of the TM model differ significantly from those of the EB model.
These differences can be explained by examining the different modes of vibration. When
co>co^ (in the regime of Case 1), the vibrating motion of the TM model is dominated by the pure
shearing motion [7, 17], and hence the EB model, which neglects the rotary inertia and pure
shear effects, become inaccurate at high frequency. As discussed in Ref. [7], at the finite cutoff
frequency, the TM shaft experiences no transverse displacement, and the cross-section of the
shaft simply rotates back and forth in unison
In Figs. 3(d)-(f) and 4(d)-(f), for = 0 and £ = 0 , the reflection and transmission coefficients
of the EB model are independent of the frequency. This is because from Eqn. (10a), A = 0, and
Eqns. (9c, d) lead to a single wavenumber F, = Tj . From Appendix I, Eqns. (32*-36*), the r and
t are thus constant matrices. It is also seen that the wave reflection and transmission coefficients
1303
for both shaft models are basically independent of the rotation speed over the entire frequency
range, even at high rotation speed = 0.05 = 44,600 rpm. In Ref. [7], it is also found that has
negligible effects on the system frequency spectrum, phase velocity and group velocity. On the
other hand, the effects of the axial load are significant for both propagating and attenuating
waves in the regime of Case H, see Figs. 3(b)-(c) and 4(b)-(c). For both shaft models under
simple support and compressive loads (Figs. 3(b, e)), the reflection coefficient ru of the incident
propagating wave is reduced significantly in the regime of Case II, while the transmission
coefficient fj, of the propagating wave component increases to balance the energy carried in the
wave. However, the attenuating wave component which does not carry any energy loses its
transmissibility in the same amount as the reflection coefficient r,2 . Thus, in the presence of a
compressive load, most of the transmitted wave energy in Case II comes from the propagating
component of the incident wave. Note that axial tensile loads have the reverse effects on these
wave components. In the clamped support case, the positive propagating wave component rn is
constant over the regime of Case II under any loading conditions for the both shaft models, as
seen in Fig. 4.
Since there is no damping at the support, the incident power (Ilinc), reflected power (Hrefi) and
transmitted power (Iltnin) in Cases II and IV are related by flinc = rirefl+ritran = Hinc,
or Iriil^+Uiil^ = 1. This relationship is confirmed by the plots shown in Figs. 3 and 4, where for
both shaft models, Irni and knl cannot exceed one. However in the regime of Case I, in which
all wave components propagate, the energy balance is Hinc = (l^ii+r2iP+lfii+r2!p) Ilinc, or Hinc =
(Iri2+r22l^+l?i2+r22l^) Hinc- Together with the plots on the phase of these coefficients (not shown
to minimize the size of this manuscript), the above relationships can also be verified for wave
motion of Case I.
1304
support without “resistance”. The impedance mismatching (rn = 1, ?,, = 0) frequency at which
the propagating wave component is completely reflected without being transmitted can also be
determined from Figs. 4(b, d) for the two shaft models. This impedance mismatching frequency
is located in the regime of Case II for the TM model where the transverse mode dominates the
vibrating motion of the shaft. Numerical results show that, as the spring constant increases, this
impedance mismatching frequency increases, but is limited to within the regime of Case II and
can never be found in the regime of Case I where the pure shearing mode dominates the vibrating
motion of the shaft (refer to Fig. 3 for the transition of types of wave motion).
Figure 6 shows the reflection and transmission coefficients for waves incident upon a support
having both translational and rotational constraints. Since both flexural and shearing modes of
vibration are constrained at this support, the maximum of the reflection coefficient is expected to
be higher than the previous case. Figures 6(a-b) and (c-d) are the results for the TM and EB
models, respectively. The translational and rotational spring constants used in the simulations
are k,Q- 10^ N/m and ^^5= 10^ Nm/rad, respectively. It is noted that in the regime of Case II, i.e..
Figure 6. Wave reflection and transmission coefficients at an elastic support with translational and rotational springs
{k,-k,Q, k^^O, c, = c, = m= y,„ = 0) as a function of frequency, J3 = 0.05 and e = 0. (a-b) and (c-d) are results
for the Timoshenko and Euler-Bernoulli shaft models, respectively.
modulus modulus
2.0
r,2 (upper line)
r,2=:r2, (middle dashed line)
Tj, (lower line)
Figure 4. Wave reflection coefficients at a clamped support (/c, = fc, = ■» and = c, = m = = 0) as a function of
frequency, (a)-(c) and (d)-(f) are the results for the Timoshenko and Euler-BernouUi shaft models, respectively. The
transition from one type of ivave motion to another is marked for the case /J = 0.05 , £ = -0.05 .
3.2, Wave reflection and transmission at elastic supports
Figure 5 shows the reflection and transmission coefficients for waves incident upon a support
with a finite translational spring for three different spring constants. Figures 5(a)-(b) and (c)-(d)
are results for the TM and EB models, respectively. The spring constant used, k^Q = 10^ N/m, is
a typical bearing spring constant value for turbine generators. The plots show that there is no
significant difference in the moduli between the two shaft models. This is because the incident
wave does not experience any rotational constraint at the support, and hence the additional rotary
inertia factor in the TM model has only a small contribution to the wave motions. As the support
spring constant increases, the curves for both the reflection and transmission coefficients are
shifted to the right and, as the spring constant approaches infinity, these curves eventually
become asymptotic to those shown in Fig. 3. Note that an impedance matching (r = 0, t = I),
where all wave components are transmitted without being reflected, is found in the high
frequency region for both shaft models. Thus, as the frequency increases, the characteristics of
waves travelling along the shaft remain unchanged such that waves propagate through this elastic
Figure 5. Wave reflection and transmission coefficients at an elastic support with a translational spring
= c, = c, =m = =0) as a function of frequency, /3 = 0.05 and £ = 0. (a-b) and (c-d) are results for the
Timoshenko and Euler-Bernoulli shaft models, respectively.
1308
in the low frequency range, both shaft models have similar reflection characteristics, and both the
reflection and transmission coefficients are not significantly affected by the rotational spring.
However, as the frequency increases, the effect of the rotational constraint on the wave motion
becomes eminent, particularly for the TM model. As seen in Figs. 6(a) and 6(c), the reflection of
the attenuating wave components are significantly higher than those of the propagating wave
components. Hence, when a rotating shaft has a clamped support(s) such as a journal bearing,
contributions from the attenuating wave components should be included in the formulation since
a significant amount of energy in the propagating component arises from the incident attenuating
wave component. It is noted that the impedance matching regions seen in Figs 5(a, c) disappear
when the rotational constraint is added. Moreover, the impedance mismatching frequency shown
in Figs. 5(b, d), which is found in the regime of Case II, also does not occur. At low frequency in
Figs. 6 (b, d), there appears to be a mismatching region, but rj, is not exactly equal to zero.
From Figs. 6(a, c), it is seen that there is a frequency at which the positive propagating wave
component r,, is zero (this frequency is slightly different for the two models). This frequency
does not correspond to an impedance matching, though the propagating wave is not reflected at
all but is only transmitted (r,i = 1). Based on other research results [18], this phenomenon likely
indicates a structural mode delocalization in bi-coupled systems, in which vibrations on both
sides of the support become strongly coupled. Further research on the vibrations of rotating
shafts with intermediate supports is being pursued to confirm the mode delocalization.
Figure 7 plots the effects of axial compressive loads on the wave reflection and transmission
upon a support with finite spring constant for the Timoshenko shaft model. As seen in Fig. 7(a),
the reflection coefficient for the incident propagating wave component ru is substantially
reduced in the low frequency range while the reflection coefficient for the incident attenuating
wave component increases significantly. However, Fig. 7(b) shows the reversed effects on the
transmission coefficient. It can therefore be concluded that, when the shaft is axially strained by
Figure 7. Wave reflection and transmission coefficients at an elastic support (A:^ = k,Q and kr = c, = Cr = m = J„, = 0)
for the Timoshenko shaft model with and without the compressive load, (a) reflection coefficients, (b) transmission
coefficients.
1309
Figure 8. Wave reflection and transmission coefficients at an elastic support k^ = and c,= Cr = m =
J„, = 0) for the Timoshenko shaft model with and without the compressive load, (a) reflection coefficients, (b)
transmission coefficients.
compressive loads, the energy contribution from the incident attenuating wave component to the
energy in the reflected propagating wave is more significant than the strain-free situation in the
low frequency range, while most of the energy in the transmitted wave derives from the incident
propagating wave component.
Figure 8 plots the wave reflection and transmission coefficients along an axially compressed
Timoshenko shaft model at a support with finite translational and rotational spring constants.
Similar results to the previous example can be observed in terms of energy contribution from the
incident attenuating wave component in the low frequency range. However, the effects of the
axial compressive load on both the reflection and transmission coefficients for the propagating
wave component ( r^^ and r,, ) are significantly reduced when compared to Fig. 7.
3.3. Wave reflection and transmission at damped supports
Figure 9 shows the effects of both translational and rotational dampers at a support with finite
translational and rotational spring constants. Figures 9(a, b) and (c, d) are results for the TM and
EB models, respectively. The translational and rotational damping constants used in this study
are c^to = 2x10^ Ns/m and Cdro = 64x10^ N-m-s/rad, typical values for bearings in turbine
generators. The curves with symbols (• and ♦) are the results when the rotational damping factor
is also included in the formulation. It can be seen that I and Ir^l for both shaft models are
significantly lowered due to the presence of damping. Note that, because of the damping, the
frequency at which 1 rj, I = 0 (compare with Figs. 6(a, c)) no longer exists for both shaft models. It
can also be seen that the effect of the rotational damping factor on the wave reflection and
transmission is not significant over the entire frequency range for both shaft models. For TM
model, the contribution of the rotational damping to both Ir;, 1 and 1?,, I is almost negligible. The
1310
support condition considered in this particular example is simulated as an actual bearing support
adopted in turbine generators. Hence for this particular type of bearing support, the effect of the
rotational damping on wave reflection and transmission is not considerable. Other numerical
results (not shown in this paper) show that the wave propagation at the damped support is
characterized by translational damping rather than rotational damping. Note that similar results
have been presented for the support without damping (see Fig. 6).
Figure 9. Wave reflection and transmission coefficients at an elastic support with damping (k, = k,o, K = Ko, c, = c,;,o,
c, = cjro and m = J,„ = 0) as a function of frequency for /3 = 0.05 and £ = 0. (a-b) and (c-d) are results for the
Timoshenko and Euler-Bernoulli shaft models, respectively.
3.4. Wave reflection and transmission at a rotor mass
Consider a gear rigidly assembled to a rotating shaft. The gear is assumed to be perfectly
balanced and its thickness is sufficiently small such that wave reflection and transmission due to
the geometric discontinuity between the shaft and the gear can be neglected. However the gear
does resist the translational and rotational motions of the cross-sectional element of the shaft.
1311
Figure 10 shows the reflection and transmission upon the gear when the mass mo and mass
moment of inertia J^o of the gear are 4 and 16 times of the shaft, respectively. Not shown in
Figs. 10 (b, d) is that r,2 = 0 when w= /„ = 0. Like some previous support conditions
discussed, the effects of the rotor mass are much more significant in the high frequency region
for both models (particularly around and beyond the cutoff frequency for the TM model). In
general, the rotor mass decreases the transmission and increases the reflection of the wave. At
very high frequency, there is basically no wave transmission. Note that, since the geometric
discontinuity between the shaft and the gear is neglected in this model, one may expect that the
actual reflection for both the propagating and attenuating wave components would be higher.
(a)
(m= = V
(b)
- (m = 0,
- ( m = mg . )
/
Figure 10. Wave reflection and transmission at a rotor mass assembled to a rotating shaft (*, = 0 = L = Cf = Cr = 0,
and m = rriQ, and J,„ = J„^) as a function of frequency when P = 0.05 and £ = 0. (a-b) and (c-d) are results for the
Timoshenko and Euler-Bernoulli shaft models, respectively.
4. WAVE REFLECTION AND TRANSMISSION AT A GEOMETRIC DISCONTINUITY
It is common for a rotating shaft element to have changes in cross-section, or to be joined to
1312
Case I, II, or IV
A, = (1 + a)co- - 2j3co - 16£ (1 + £ - ~)
a
Z=0
Case I, II, or IV
B, = co'
ccco- - 2p(o - I6a (1 + £)(1 + £ - )
a
A={\ + a)(o^-2^co-^^^ (l + £, -— )
o" a
16a
= co^\ a (O' -215(0-^^ (l+£j(l + £ -— )
L O’ a
Figure 11. Wave reflection and transmission at a geometric discontinuity.
another shaft element by a coupling. Figure 1 1 shows a typical example of a discontinuous shaft
model in which two shafts of differing wavenumber and diameter are joined at z = 0 . The
subscripts I and r denote z = O' and z = 0^ regions, respectively. It is known that when a wave
encounters a junction or a discontinuity, its wavenumber is changed. It is therefore possible that
a wave on the left side of the junction can be propagating, while after crossing the junction to the
right side, the wave becomes attenuating. Therefore, for a Timoshenko shaft, when a wave
propagates through the junction, there are mathematically nine possible different combinations of
wave motions to be considered depending on the values of the functions A and B on each side of
the junction, as depicted in Fig. 12.
Figure 12. Nine possible combinations of wave motions at a geometric discontinuity of the cross section for the
Timoshenko shaft model. Subscripts / and r denote the left and the right side of the discontinuity, respectively.
1313
For simplicity, assume that material properties such as p, E, and G are the same for both sides
of shaft element. The displacement continuity, moment and force equilibrium conditions are
applied at the junction to determine the wave refection and transmission matrices. Results for the
three most commonly encountered possibilities in the low frequency regime are listed as follows.
Case II {A, >0, Bi <0) - Case I (A^ > 0 , R, >0):
■ 1 r
1
1 '
r 1 M
rC" =
tCY
Jlu ^21 .
nu ■
~n2i.
Jl\r ^2r_
r -i^Bu
^21^2!
C" +
r Xtnu
”Y,r72, 1
L^xr.,
-riu) r
21 -in 21^
1
Y
rr
1
T
^772,)J
rC-^
Yuriu 72rn2r
L-io-"(r„-77„) -i<y^(r2r-V2r)J
tc",
Case II (A, >0, B, <0) - Case II (A^ > 0 , B^<0):
■ 1 1 ■
■ 1 1 ■
■ 1 1 *
C" +
rC^ =
Jlv n2l.
— nu “^2/.
Y.r n2r.
(37a)
(37b)
(38a)
-"^21^21
^2, -in 21
C" +
i^unu ^21^21
~i(^H — nu ) ~(^2/ ~ ^^21 )
rC"
-i<yXrnu -^Xr'n2r
-zcr"(r,,-77„) -C7“(r2,-7]2,)J
CaseII{A^>0, B, <0) - Case IV {A, <0 , S, <0):
r i 1 1
r 1 1 1
■ 1 r
C" +
rC^ =
nu n2i
l-nu -n2i^
n2r n^r.
Y/^2/
^2/^2/
JXi—nu) Y/~^*^2;.
“(Yf -in2i)_
-i^%rn2r
-<yXrn:r
tc\
(38b)
(39a)
(39b)
where cr is the diameter ratio between the shaft elements, defined as
1314
(40)
CJ = — .
Note that 77/5 in Eqns. (37a-39b) are given by Eqns. (20a, b), (22a, b), and (24a, b) according to
the type of wave motion, and 77/ s on the right side of the geometric discontinuity are modified
as follows.
where
: - 77 = - — for Case I,
r„e; yzX
(41a, b)
-co^ Tl+co-
, Ti, - — - 7- for Case II,
r„e; iT^rS;
(42a, b)
,7]2 for Case IV,
ir,x r,,e;
(43a, b)
£' = 1 + 8,. — and £, = — .
(44)
a O'
Moreover, the wavenumbers, A and B of the shaft element on the right side of the junction are
modified as follows.
y „ = ^[a, + ^Aj-4B^f . r,, = - Va?-4B,)^ , (45a. b)
+ = . (45c, d)
where,
A. = (l + a)a.= -2pa)-^a + £,-|),
B. =co‘
am--2Pa-^ (l + £,)(l + £,-^)
C OL
(46b)
Corresponding results for the simple Euler-Bemoulli shaft model are listed in Appendix II.
Figures 13 to 16 show some representative examples of wave reflection and transmission
upon the geometric discontinuity. In Figs. 13 and 14, the thick and thin curves represent results
1315
for the TM and EB models, respectively. The second graph in each figure shows the changes of
Ai, Bi, Ar and Br, and how wave solutions on both sides of the discontinuity change as the
frequency increases for the TM model. In general, the wave reflection and transmission for the
EB model are frequency independent except when the shaft is axially strained, while the wave
propagation characteristics for the TM model are strongly dependent on the frequency.
Comparing Figs. 13 and 14, it is noted that, for both shaft models, the average reflection and
transmission rates for cr = 0.8 are higher than those for o' = 1.2 , especially for the attenuating
wave components. These results imply that incident attenuating waves contribute more energy to
propagating waves at the discontinuity when the waves travel from a smaller to a larger cross-
section. In particular, it is noted that the transmissibiiity of the attenuating wave tn has a strong
dependency on the direction of propagation. Note also that the differences between the two shaft
models are more pronounced when cr = 0.8 . It is clearly seen from the figures that when Bj and
B^ change from negative to positive, both reflection and transmission coefficients experience a
sharp jump or drop at the finite cutoff frequencies, due to changes in the types of wave motion.
In the frequency region (S, > 0 and B^ <0) located between the two cutoff frequencies in Fig.
2.0 h
0.5
Thick curves : Timoshenko shaft model
Thin curves : Euier-Bernoulli shaft model
- Ui
h
i
il _
-
7/ X. - V'
.'7
\
5
(O
Figure 13. Reflection and transmission of waves incident upon a change in the cross-section, a= 0.8, P = 0.05, and
£ = 0. Thick and thin curves are results for the Timoshenko and Euier-Bernoulli shaft models, respectively. Note
that the ordinates in the lower graphs keep increasing with frequency (abscissa).
1316
13, the wave motion on the left side of the junction is governed by the wave solution of Case I
since all wave components are propagating at a frequency larger than the cutoff frequency, while
the wave motion on the right side of the junction is governed by the wave solution of Case II.
Thus, for O’ = 0.8 , some of the propagating wave components on the left side of the shaft element
cannot propagate as they pass the discontinuity, and become attenuating. A similar, but converse
conclusion can be drawn for the frequency region (5^ > 0, 5, < 0) when cr = 1.2 , as shown in
Fig. 14. The results of Figs. 13 and 14 show that, for different system parameters cr, jS, and e and
at any given frequency, the types of wave motion on each side of the discontinuity can be
different, as depicted in Fig. 12.
From Eqns. (41a-43b), it is seen that when the Timoshenko shaft is axially strained and (O is
not sufficiently large, the wavenumber (hence wave propagation characteristics) depends strongly
on the cross-section ratio <7. Figure 15 shows the effects of the axial load on the wave reflection
and transmission, which are mostly limited to the relatively low frequency region. In Figs. 15(a-
b), when the shaft is axially compressed (£ = -0.05), the reflection and transmission due to the
incident attenuating wave component decrease for both o < 1 (plot (a)) and <7 > 1 (plot (b)).
However, the transmission due to an incident propagating wave decreases significantly for
— — f, 2
2.0 h
Figure 14. Reflection and transmission of waves incident upon a change in the cross-section, cr= 1.2, /3 = 0.05, and
£ = 0. Thick and thin curves are results for the Timoshenko and Euler-Bernoulli shaft models, respectively. Note
that the ordinates in the lower graphs keep increasing with frequency (abscissa).
1317
Figure 15. Reflection and transmission of waves upon a change in the cross-section when = 0.05 for the
Timoshenko shaft model, (a) <7=0.8 and £ = -0.05. (b) (T= 1.2 and e= -0.05. (c) cr= 0.8 and £= 0.05. (d) or = 1.2
and e= 0.05. Thin and thick curves show the results when the shaft is strain-free (£= 0) and strained, respectively.
c - 0.8 and increases for <7 = 1.2 at low frequency.
Effects of the axial load on the wave reflection and transmission are more significant when
the shaft is compressed (Figs. 15(a-b)) than when it is under tension (Figs. 15(c-d)). This is
because the wavenumbers of both the propagating and attenuating wave components are only
slightly changed. It is also noted that, in the low frequency range, the wave solution of Case IV
governs the wave motions on both sides of the discontinuity, and the wave components which
have large wavenumber (Fi) attenuate, while wave components with small wavenumber (r2)
propagate along the waveguide as long as A remains negative.
1318
5. WAVE REFLECTION AT BOUNDARIES
When a wave is incident upon a boundary, it is only reflected because no waveguide exists
beyond the boundary. Consider an arbitrary boundary condition with translational and rotational
spring constraints, dampers, and a rotor mass, as shown in Fig. 16. The reflection matrix at the
boundary is derived for each Case. Applying the same non-dimensional parameters employed in
Section 3, and by imposing the force and moment balances at the boundary, which can be
deduced by eliminating and inEqns. (27a, b),
M~ = k^y/ + c^yr + J„y/ , (47 a)
-V = k,u + c,u + mu , (47b)
the reflection matrix for each Case is determined.
CaseI{A>0, B>0):
J r72(ir2-2:,J T’r-T7,(jri + ^.) +
+ Kr2-Bi) + ^s\ Kr2-T72)-^.J’
Case 7/ ( A > 0 , 5 < 0):
7],(ir,-2:j T'r-77,(ir,+Ej -7]2(r2 + zj1
'''[i(r,-r7,)+z, (r,-ii7,)+i.J [i(r,-j?,)-z,
M,Jm
Z=0
Figure 16. Wave reflection upon a general boundary.
1319
CaseIV{A<0, 5<0):
772 (^r2 s„,)
iCFj - 7)2) +
(r.-mj+s.
772 (zTj + Sp,)
/(r2 ~ ^2) ■“
-^iCr.+zj
(r,-j77,)-E,
(50)
where 77's in above equations have been defined in Eqns. (20a, b), (22a, b) and (24a, b), and
Z„, = + ic^co - J,„co~ , and = ^, + ic,Q) - mco^ , (51a, b)
The corresponding results for the simple Euler-Bernoulli shaft model are listed in the Appendix
in. By specifying the parameters in the reflection matrix r, results for three typical boundary
conditions (simple support, clamped support, and free end) can be obtained.
• Simple support {k, =^, =m = c, = = 7,„ = 0)
r
-1 0
0 -1
for Case I, II, and IV,
(52)
• Clamped support (k^ = k,. = m = c, = = J„, = 0)
1
Tli-ri2
(53b)
(53c)
• Free end {k, = k^ = m = c, = = /„, = 0 , and £ = 0)
1
rii+ri2
2772
77,- 772 L-277i -(r?, +772)J
1
-irii+Tli) "2772
2t}, (771+772)
for Case II,
for Case IV,
■77, +7?2 2772
-277, -(771+772),
for Case I,
(53a)
r
J_
77,7?2(ri+72)”7ir2(77, +772)
-277,7i(77, -7i)
277272(772-72)
-77,772(71 +72) + 7i72(77i +772).
(54a)
where. A, = r],n, - 7, ) + 7,72 (77, - ^2 ) for Case I,
+72) + 7i72(77i +772) 2i7}^y^iin^ -y^)
^ A;, [ 2/77,7,(771 -7,) 77,772(/7, +72)-7i72(7?, +772)
(54b)
where, =77,772 (77, -72) -7,72(7?, - 772) for Case//,
1320
Figure 17. An example of a rotating shaft with multiple supports and discontinuities.
I \TiiV2(ri+ir2)-rj2(Tii+r]2) 277,7,(77, +ir,) ' ^
r = - (54c)
A/v L -2i7?2r,(77j-7j -77,772(7, +J72)+rir2(^i +^2).
where, A;^ =77,772(71 -172) “7172 (Hi -772) for Case IV.
6. APPLICATIONS
The reflection and transmission matrices for waves incident upon a general point support or a
change in cross-section can be combined with the transfer matrix method to analyze the free
vibration of a rotating Timoshenko shaft with multiple supports and discontinuities, and general
boundary conditions. The basic idea of this technique has been shown in Ref. [6]. However, due
to the complex wave motions in the Timoshenko shaft model, such as the frequency dependency
of the wave reflection and transmission at a cross-section change, it is important to apply the
proper reflection and transmission matrices consistent with the values of A and B on both sides of
the discontinuity, particularly when numerical calculations are performed. Consider for example
the free vibration problem of the rotating Timoshenko shaft model shown in Fig. 17. Denoting R
as a reflection matrix which relates the amplitudes of negative and positive travelling waves at a
discontinuity, and defining T/ as the field transfer matrix which relates the wave amplitudes by
C^iZo + z) = TC"(Zo) , C-(zo + z) = T-'C-(zo) , (55)
the following relations can be found.
1321
W-=R5W^
(R5=>-s).
(56a)
i = 2,3,4 (station number),
/i = left (/) or right (r)
(56b)
w>T,w-,,
(56c)
< =r,w-,
(56d)
<=T.w;,
(56e)
where in Eqn. (56b),
R,,=TiR„,,T,. R,,=r,+t,(R-’-r,-)-'t, (56f)
Solving the above matrix equations gives
(r,T,R„T,-I)w>0, (57)
where each element of the matrix is a function of two different wavenumbers and the frequency
CO. For non-trivial solutions, the natural frequencies are obtained from the characteristic equation
Det[(r,T,R„T,-I)] = 0. (58)
The proposed method is applied to an example of a two-span rotating shaft, simply supported
at the ends and with an intermediate support consisting of translational and rotational springs, as
shown in Fig. 18. Numerical computations were performed by a PC-based Mathematical. The
values of the spring constants Kj and Kr are those introduced in Section 3.2, with ^ = 1 m and
the rotation parameter p = 0.05 .
1322
Figure 19 shows the first eight natural frequencies of the vibrating shaft for both the TM and
EB models for a classical simple intermediate fixed support K^=0) placed at
various locations. The results confirm the well-known fact that the Timoshenko model leads to
smaller eigenvalues. Figure 20 shows the first eight natural frequencies of the Timoshenko shaft
for an elastic intermediate support with three different translational spring constants. It can be
seen that the effect of the translational spring diminishes for higher modes. The proposed wave
analysis technique can also be applied effectively to the study of structural mode localizations in
mistuned, rotating systems. Dynamics of such systems will be addressed in another paper.
Figure 19. Natural frequencies of a two-span, rotating
shaft as a function of the support location; intermediate
support is fixed.
Figure 20. Natural frequencies of a two-span rotating
Timoshenko shaft as a function of the support location;
intermediate support consists of k, and kr .
7. SUMMARY AND CONCLUSIONS
In modern high speed rotating shaft applications, it is common that the shaft has multiple
intermediate supports and discontinuities such as bearings, rotor masses, and changes in cross-
1323
sections. In many cases, the ratio of the shaft diameter to its length between consecutive supports
is large, and the Timoshenko model (TM) is needed to accurately account for the shear and rotary
inertia effects. In this paper, the wave propagation in a rotating, axially strained Timoshenko
shaft model with multiple discontinuities is examined. The effect of the static axial deformation
due to an axial load is also included in the model. Based on results from Ref. [7], there are four
possible types of wave motions {Cases I, II, III and IV) in the Timoshenko shaft, as shown by
Eqns. (5a-8b). In practice. Case III does not occur and is excluded in the analysis. For each
Case, the wave reflection and transmission matrices are derived for a shaft under various support
and boundary conditions. Results are compared with those obtained by using the simple Euler-
Bemoulli model (EB) and are summarized as follows.
1) In general, the two shaft models show good agreement in the low frequency range where the
wave motion is governed by Case II and Case IV. However, at high frequencies, the types of
wave motions and propagation characteristics for the TM and EB models are very different.
2) The effects of shaft rotation on the wave reflection and transmission are negligible over the
entire frequency range and even at high speed (up to 44,600 rpm). While the effects of the
axial load are significant, especially in the low frequency range.
3) When waves are incident at supports with only translational springs, differences in the results
between the TM and EB models are small, and there exists frequency regions of impedance
matching and an impedance mismatching frequency (limited to within the regime of Case IT).
The impedance matching and mismatching disappear when a rotational spring is added to the
support. Instead, there is a frequency at which Irul = 0 and Ifni = 1, and vibrations on both
sides of the support become strongly coupled. This (delocalization) phenomenon suggests
further research on the vibrations of constrained multi-span beams. When there is damping
at the support, the frequency at which IrnI = 0 does not occur. Moreover, effects of
translational damping on the wave propagation are more significant at high frequency,
especially for the TM model, however effects of rotational damping is not significant over the
entire frequency range.
4) Contributions of attenuating wave components to the energy in the reflected and transmitted
waves are significant when the shaft is axially strained and when the support has a rotational
constraint. Thus attenuating waves should be included in the formulation.
5) Unlike the spring supports, in which waves are easily transmitted at high frequency, the rotor
mass support diminishes the wave transmission as the frequency increases.
6) When waves are incident at a geometric discontinuity such as a change in the cross-section,
there are nine possible combinations of wave motions on both sides of the discontinuity. It is
shown that differences of the results between the TM and EB models depend on the diameter
ratio (and hence the direction of the wave incidence). Moreover, incident attenuating waves
contribute more energy to propagating waves at the discontinuity when the waves travel from
a smaller to a larger cross-section. When the shaft is axially strained, the effects of the load
on the wave propagation are primarily limited to the low frequency range.
The reflection and transmission matrices are combined with the transfer matrix method to
provide a systematic solution method to analyze the free vibration of a multi-span, rotating shaft.
Since the procedure involves only 2x2 matrices (while including the near-field effects already),
strenuous computations associated with large-order matrices are eliminated.
1324
ACKNOWLEDGMENTS
The authors wish to acknowledge the support of the National Science Foundation and the
Institute of Manufacturing Research of Wayne State University for this research work.
REFERENCES
1. Lin, Y.K., Free Vibrations of a Continuous Beam on Elastic Supports. International Journal
of Mechanical Sciences, 1962, 4, pp. 409-423.
2. Graff, K.F., Wave Motion in Elastic Solids, Ohio State University Press, 1975.
3. Cremer, L,, Heckl, M. and Ungar E.E., Structure-Bome Sound, Springer-Verlag, Berlin,
1973.
4. Fahy, F., Sound and Structural Vibration, Academic Press, 1985.
5. Mead, D.J., Waves and Modes in Finite Beams: Application of the Phase-Closure Principle.
Journal of Sound and Vibration, 1994, 171, pp. 695-702.
6. Mace, B.R., Wave Reflection and Transmission in Beams. Journal of Sound and Vibration,
1984, 97, pp. 237-246.
7. Kang, B. and Tan, C.A., Elastic Wave Motions in an Axially Strained, Infinitely Long
Rotating Timoshenko Shaft. Journal of Sound and Vibration (submitted), 1997.
8. Han, R.P.S. and Zu, J.W.-Z., Modal Analysis of Rotating Shafts: A Body-Fixed Axis
Formulation Approach. Journal of Sound and Vibration, 1992, 156, pp. 1-16.
9. Zu, J.W.-Z. and Han, R.P.S. , Natural Frequencies and Normal Modes of a Spinning
Timoshenko Beam With General Boundary Conditions. Transactions of the American
Society of Mechanical Engineers, Journal of Applied Mechanics, 1992, 59, pp. 197-204.
10. Katz, R., Lee, C.W., Ulsoy, A.G. and Scott, R.A., The Dynamic Response of a Rotating
Shaft Subject to a Moving Load. Journal of Sound and Vibration, 1988, 122, pp. 131-148.
11. Tan, C.A. and Kuang, W., Vibration of a Rotating Discontinuous Shaft by the Distributed
Transfer Function Method. Journal of Sound and Vibration, 1995, 183, pp. 451-474.
12. Argento, A. and Scott, R.A., Elastic Wave Propagation in a Timoshenko Beam Spinning
about Its Longitudinal Axis. Wave Motion, 1995, 21, pp. 67-74.
13. Dimentberg, F.M., Flexural Vibrations of Rotating Shafts, Butterworth, London, 1961.
14. Dimarogonas, A.D. and Paipeties, S.A., Analytical Method in Rotor Dynamics, Applied
Science, New York, 1983.
15. Lee, C.W., Vibration Analysis of Rotors, Kluwer Academic Publishers, 1993.
16. Choi, S.H., Pierre, C. and Ulsoy, A.G., Consistent Modeling of Rotating Timoshenko Shafts
Subject to Axial Loads. Journal of Vibration and Acoustics, 1992, 114, pp. 249-259.
17. Bhashyam, G.R. and Prathap, G., The Second Frequency Spectrum of Timoshenko Beams.
1325
Journal of Sound and Vibration, 1981, 76, pp. 407-420.
18. Riedel, C.H. and Tan, C. A., Mode Localization and Delocalization of Constrained Strings
and Beams. Proceedings ofASME Biennial Conference on Mechanical Vibration and Noise
(submitted), 1997
1326
For simple and clamped supports, the reflection and transmission matrices are listed as follows.
Simple support ( k. = k. = n
l = C,
= c, = .
Case // ( A > 0 , B < 0):
1
■ r.
r, ■
iF, r2
'ir,
r./
'"ir.-r2
.-ir^
-r^.
CaseIV{A<0, B<0):
1
-ire
1 1
t “
'r.
-ir:
iT, +r2
.-^2
-n_
i ^T,+r2
-r2
• Clamped support {k^ =<=<>, m=c, = c, = J„, = 0 ); t = 0 .
CaseII{A>0, B<0):
1 ^^1+^2 2r2
-2iT, -(ir, + r,)J’
CaseIV{A<0, B<0):
1 r-(ir,-r2) -2iTi '
^“iT.+r.L -2r2 iT, -r^ ■
(32a*, b*)
(33a*, b*)
(35*)
(36*)
APPENDIX II
The reflection and transmission matrices for a wave incident upon a cross-sectional change
for the simple Euler-Bernoulli shaft model can be determined by solving the following sets of
matrix equations. Only two representative combinations are shown.
CaseII(Ai>0, B, <0)- Case II {A,>0, B,<0):
1327
Casen{A,>0, <Q) - Case TV {A, <0 , <0):
1
1
1
1 1 .
1
1 1 .
■ +
-
•^2,.
-F^ F"
^1/ •*■2/
C*
+
F2 1
rC" =
[T^ _r3
L “ 1/ ^ 21
2/ J
r,rj
where, F sr and T2r have been defined in Eqns. (45c, d), and A^- and Br are given by
O’"
G
(39a')
(39b')
(46a')
(46b*)
If the rotating shaft is strain-free, then r can be reduced to simple forms representing typical
boundary conditions such as simple support, clamped support, and free end as shown in Ref. [6].
Note that for those supports in the strain-free case, the reflection matrices are constant.
1328
ANALYTICAL MODELLING OF COUPLED VIBRATIONS OF
ELASTICALLY SUPPORTED CHANNELS
Yavuz YAMAN
Department of Aeronautical Engineering, Middle East Technical University
0653 1 Ankara, Turkey
An exact analytical method is presented for the analysis of forced vibrations
of uniform thickness, open-section channels which are elastically supported at
their ends. The centroids and the shear centers of the channel cross-sections do
not coincide; hence the flexural and the torsional vibrations are coupled. Ends
of the channels are constrained with springs which provide finite transverse,
rotational and torsional stiffnesses. During the analysis, excitation is taken in
the form of a point harmonic force and the channels are assumed to be of type
Euler-Bernoulli beam with St.Venant torsion and torsional warping stiffness.
The study uses the wave propagation approach in constructing the analytical
model. Both uncoupled and double coupling analyses are performed. Various
response and mode shape curves are presented.
1. INTRODUCTION
Open-section channels are widely used in aeronautical structures as stiffeners.
These are usually made of beams in which the centroids of the cross-section
and the shear centers do not coincide. This, inevitably leads to the coupling of
possible flexural and torsional vibrations. If the channels are symmetric with
respect to an axis, the flexural vibrations in one direction and the torsional
vibrations are coupled. The flexural vibrations in mutually perpendicular
direction occur independently. In the context of this study, this type of
coupling is referred to as double-coupling. If there is no cross-sectional
symmetry, all the flexural and torsional vibrations are coupled. This is called
as triple-coupling. The coupling mechanism alters the otherwise uncoupled
response characteristics of the structure to a great extent.
This problem have intrigued the scientists for long time. Gere et al [1], Lin
[2], Dokumaci [3] and Bishop et al [4] developed exact analytical models for
the determination of coupled vibration characteristics. All those works, though
pioneering in nature, basically aimed to determine the free vibration
characteristics of open-section channels.
The method proposed by Cremer et al [5] allowed the determination of
forced vibration characteristics, provided that the structure is uniform in
cross-section. The use of that method was found to be extremely useful when
the responses of uniform structures to point harmonic forces or line harmonic
loads were calculated. Mead and Yaman presented analytical models for the
1329
analysis of forced vibrations of Euler-Bernoulli beams [6]. In that they
considered finite length beams , being periodic or non-periodic, and studied
the effects of various classical or non-classical boundary conditions on the
flexural response. Yaman in [7] developed mathematical models for the
analysis of the infinite and periodic beams, periodic or non-periodic Kirchoff
plates and three-layered, highly damped sandwich plates.
Yaman in [8] also developed analytical models for the coupled vibration
analysis of doubly and triply coupled channels having classical end boundary
conditions. In that the coupled vibration characteristics are expressed in
terms of the coupled wave numbers of the structures. The structures are first
assumed to be infinite in length, and hence the displacements due to external
forcing(s) are formulated. The displacements due to the waves reflected from
the ends of the finite structure are also separately determined. Through the
superposition of these two, a displacement field is proposed. The application
of the end boundary conditions gives the unknowns of the model. The
analytical method yields a matrix equation of unknowns which is to be solved
numerically. The order of the matrix equation varies depending on the number
of coupled waves. If the cross-section is symmetric with respect to an axis
( double-coupling) and if the warping constraint is neglected, the order is six.
If there is no cross-sectional symmetry (triple-coupling) and if one also
includes the effects of warping constraint, the order then becomes twelve.
This order is independent of the number of externally applied point forces.
Although the method is basically intended to calculate the forced response
characteristics, it conveniently allows the computation of free vibration
characteristics as well. The velocity or acceleration of a point can easily be
found. The mode shapes can also be determined. Both undamped and damped
analyses can be undertaken.
This study is based on the models developed in reference [8] and aims to
analyze the effects of non-classical end boundary conditions on the coupled
vibratory responses. If the ends are elastically supported (which may also have
inertial properties) the problem becomes so tedious to tackle through the
means of classical analytical approaches. The current method alleviates the
difficulties encountered in the consideration of complex end boundary
conditions.
In this study a typical channel, assumed to be of type Euler-Bernoulli beam, is
analyzed. It represents the double-coupling. Effects of the elastic end
boundary conditions on the resonance frequencies, response levels and mode
shapes are analyzed. Characteristics of otherwise uncoupled vibrations are
also shown.
1330
2. THEORY
2.1 Flexural Wave Propagation in Uniform Euler-Bernoulli Beams
Consider a uniform Euler-Bernoulli beam of length L which is subjected to a
harmonically varying point force Fo e acting at x=Xf. The total flexural
displacement of the beam at any Xr (0 < Xr < L) can be found to be [5-8],
w(x,,t) = ( i;A„e''„\ + Fo i ane'^'
r r
)e'
(1)
The first series of the equation represents the effects of four waves which are
being reflected from the ends of the finite beam. They are called free-waves.
The second series accounts for the waves which are being created by the
application of the external force Fo e on the infinite beam. Those waves are
known as forced-waves, kn is the n’th wave number of the beam and
kn =(mco^/EI)'^'‘ where m= Mass per unit length of the beam, co= Angular
frequency, EI= Flexural rigidity of the beam, an values are the complex
coefficients which are to be found by satisfying the relevant compatibiliy and
continuity conditions at the point of application of the harmonic force [6,7].
An values, on the other hand are the complex amplitudes of the free waves and
are found by satisfying the required boundary conditions at the ends of the
beam. Once determined, their substitution to equation (1) yields the flexural
displacement at any point on the finite beam due to a transversely applied
point harmonic force. More comprehensive information can be found in [7].
2.2 Torsional Wave Propagation in Uniform Bars
If one requires to determine the torsional displacements generated by a point,
harmonically varying torque, a similar approach to the one given in Section
2.1 can be used. In that case, the total torsional displacement can be written as:
= )e‘”' (2)
k is the wave number of the purely torsional wave and is known to be
k=(-pIoa)VGJ)^^^ . k2= -ki and GJ=Torsional rigidity of the beam, p=Material
density, Io=Polar second moment of area of the cross-section with respect to
the shear centre. Toe'“ ^ is the external harmonic torque applied at x=Xt and
b=l/(2kGJ). Bn values are the complex amplitudes of the torsional free-waves
and are found by satisfying the appropriate end torsional boundary conditions.
The consideration of the warping constraint To modifies equation (2) to the
following form.
1331
(3)
«x„t) = (£C„e^\ + Toic„e-'=„'V,' ) e
n*l »•!
Now kn are the roots of
EFo kn'^-GJkn^-pIoCO^=0
(4)
Cn values are found by satisfying the necessary equilibrium and compatibility
conditions at the point of application of the point harmonic torque acting on an
infinite bar [8]. Cn values are determined from the end torsional boundary
conditions of the finite bar.
2.3 End Boundary Conditions for Uncoupled Vibrations
2.3.1 Purely Flexural Vibrations
Consider an Euler-Bernoulli beam of length L which is supported by springs at
its ends. The springs provide finite transverse and rotational constraints Kt and
Kr respectively. The elastic end boundary conditions can be foimd to be:
El w”(0) - Kr I w’(0)=0 El w’”(0) + Ktj w(0)=0
El w”(L) + K r,r w’(L)=0 El w’”(L) - Kt,rW(L)=0 (5)
Here w’=
dw(x)
w
d‘w(x)
and w’”
dV(x)
w(x) is the spatially
dx ’ dx“ dx'
dependent part of equation (1) and second subscripts 1 and r allows one to use
different stiffnesses for left and right ends. A more comprehensive study on
these aspects can be found in references [6,7].
2.3.2 Purely Torsional Vibrations
Now consider a bar of length L which is supported by torsional springs,
having finite Ktor,at its ends. The elastic end boundary conditions requires that,
Torque (0) - Ktor,i (|)(0)=0 and Torque (L) + Ktor .r (i)(L)=0 (6)
Depending on the consideration of the warping constraint To, the torque has
the following forms
Torque(x)=GJ — — or Torque(x)= GJ — — - Ei o , 3 ( /)
dx dx ux
1332
(})(x) in equation (7) should be obtained either from equation (2) or equation (3)
depending on the warping constraint r©.
2.4 Doubly-Coupled Vibrations
Now, consider Figure 1 . It defines a typical open cross-section which is
synunetric with respect to y axis
V z
(b)
Figure 1 : A Typical Cross-section of Double-coupling
( a. Coordinate System, b. Real and Effective Loadings
C: Centroid, O: Shear Centre)
A transverse load applied through C results in a transverse load through O and
a twisting torque about O. In this case the flexural vibrations in z direction are
coupled with the torsional vibrations whereas the flexural vibrations in y
direction occur independently. The motion equation of the coupled vibrations
is known to be [1,2].
a-w a-(j)
= 0
a‘‘(i) 3^6 a’w a"(j)
(8)
If one assumes that,
w(x,t) = w„eVe”'
(|)(x.t) = 4>„eVe“’ (9)
1333
Then, it can be found that, a load Pz through the centroid will create the
following displacements at any x (0 < x < L) along the length of the
channel [8],
w(x,t) = (2A„e^’‘+Pjt a„ e'^ )e'"'
n^l nvl
n«l n>I
Now kn values are the coupled wave numbers, An values are the complex
amplitudes of the coupled free waves, an values are the complex coefficients
which are to be found by satisfying the required compatibility and continuity
conditions and 'Pn^ ( (El^ kn'^-mco^) / (CymcD^) ) [8].
If required, the warping displacement u(x,t) can be found from (l)(x,t )as
u(x,t)=-2A,^iM (11)
dx
where As is the swept area.
Here 2j gives the order of the motion equation. j=3 defines the case in which
the effects of warping constraint are neglected and j=4 represents the case
which includes the warping effects.
An values are found by satisfying the necessary 2j end boundary conditions. If
warping constraint is neglected, the required six boundary conditions have the
general forms given in equations (5) and (6). But the forms of w(x) and (j)(x)
are now those given by equations (10) with j=3. If the warping constraint is
included in the analysis the boundary conditions become eight. The six of
those are again found by considering equations (10) with j=4 and substituting
the resultant forms into equations (5) and (6). The remaining two can be
found by evaluating equation (1 1) at both ends.
When the flexural and torsional displacement expressions are substituted into
the relevant equations, a set of equations is obtained. For the case of a load
Pz and no warping constraint, the following equations can be found for j=3.
EI^ w’”(0) + Kt,iw(0)=0
EI^ (E kn^ An+ (-1) PzZ - kn^ a „ 6 ' "f ' )
^t,l ( E ■^n ■^PzE^n^ n f ) “0 (12)
1334
(13)
EI^w”(0) -Kr,i w’(0)=0:
EI^( I kn'An + Pz t kn'an 6 ‘ ' )
n«l n«t
■ ( S kn An + Pz^'kn^n® n f )
n*l n“l
GJ^^l«=o-K,„,.i W)=0:
dx
GJ( S k„ T „ A„ + (- 1 ) (Pz cy) t -k„ 'f „ a„ e ' ’‘f ')
fl«l 11' I
-K,or,i ( i % A„ + (Pz Cy) i >?„ a„ e ')=0
diKx) I
GJ-^Ix.l + K,„„ <KL)=0:
dx
GJ( 2k„>P„ A„ e“+ (PzCy) i-k„'P„a„ )
ns I 11=1
+ K,„,.r ( I % A„ + (Pz Cy) t Tn a„ e "f ')=0
El5W”(L)+ Kr,rW’(L)=0:
El^CS k„^A„e^‘-+(Pz) i k„^a„ )
n-l ii’*)
+ Kz,r( i k„A„ e“+ (Pz) t-k„a„ e■^'‘-^')=0
n»l n=i
(14)
(15)
(16)
EI^ w’”(L)-Kt,rW(L)=0:
Eiaz kn'Ane'n^ +(Pz) 2 - kn^ a „ e n ' ^ )
11=1 Iia]
-K,|(Z A„eV+(Pz) Za„e^''-V)=0 (17)
lt=l 11=1
Here (-1) multipliers are included due to the symmetry and anti-symmetry
effects.
Those equations can be cast into the following matrix form.
= -{Terms containing Pz} (18)
1335
An eighth order equation represents the necessary matrix equation for the
determination of An values if the warping constraint is included in the analysis.
In that case, equations (12), (13), (16) and (17) are valid with j=4. On the other
hand equations (14) and (15) should be replaced by,
(GJ^-Er„^^)L.o <l>(0)=0:
dx dx
(GJ( £ k„ % A„ + (- 1 ) (P. Cy) £ -k„ a„ e * „ ' ’‘f ') -
nol n»l
Er„( £ k„3 'f „ A„ + (- 1 ) (P^ Cy) £ -k„ = % a„ e * „ ' -f '))
it«l IIS’!
-K,„,,,(|;'P„A„+(P,Cy)i 'P„a„e-^'’‘f')=0 (19)
cr A/T^=A.
(GJ “ EFo j 3 )• x=L I^tor>r y(L) 0 .
dx dx
(GJ( Jkn'Fn A„ e’=„‘-+ (PjCy) i-kn'Pnan ' )-
n=t Ii»l
EFoC Xkn^ 'J'n A„ eV+ (PzCy) £-k„^'P„a„ ' ))
11=11 n-l
+ Kior,, ( X % A„ + (Pz Cy) X % a„ e * „ ' \ ')=0 (20)
n-] ii3|
where j=4. The remaining two equations are found by considering the warping
of the extreme ends. If the ends are free to warp the axial stress is zero, if the
ends are not to warp the axial displacements are zero at both ends. No elastic
constraints are imposed on end warping. If the left end is free to warp and the
right end is not to warp, the required boundary conditions can be shown to be;
u’(0)=0:(x kn^'Pn A„ + PzX 'i'.ian k„^e-^ ' ^') = 0 (21)
u(L)=0 :(X k„ e^‘' % A„ + PzX (-k„) '‘-"f') = 0 (22)
n=] list
All the equations can be put into the following matrix form
A.'
A:
A,
• = -{Terms containing Pz} (23)
A,
A.
A.
A.
1336
Required An values are numerically found from equations (18) or (23). Their
substitution to the appropriate forms of equations (10) and (11) yield the
required responses at any point on the beam.
3.RESULTS AND DISCUSSION
The theoretical model used in the study is shown in Figure 1 and has the
following geometric and material properties:
L=l(m), A =1.0*10"' (m\ h = 5.0*10-^(m), 15 = 4.17*10-* (m\
Cy = 15.625*10'* (m), J = 3.33* lO"' ' (m*), =7.26*10'* (m"'), p= 2700 (kg/m*),
r„ =2.85*10''* (m*), E = 7*10'“ (N/m*), G = 2.6* 10'“ (N/m*).
Structural damping for torsional vibrations is included through, complex
torsional rigidity as GJ*=GJ(l+z|3). For coupled vibrations, it is also included
through the complex flexural rigidity as El^(H-z'n).
First presented are the results for purely torsional vibrations. A bar assumed to
have the given L, p, lo, G and J values is considered. The bar is then restrained
at both ends by springs having the same torsional stiffness Ktor- A very low
damping, p=10'^, is assigned and the resonance frequencies are precisely
determined. It is found that, the introduction of a small Ktor introduces a very
low valued resonance frequency. That fundamental frequency increases with
increasing Ktor and as torsional constraint reaches to very high values, it
approaches to the fundamental natural frequency of torsionally fixed-fixed
beam. Table 1 gives the fundamental frequencies for a range of Ktor values and
Figure 2 represents the fundamental mode shapes for selected Ktor values.
Table 1: Uncoupled Fundamental Torsional Resonance Frequencies
((3=10’^,No Warping Constraint)
Ktor [N1
Frequency [Hz]
0
0.
10'^
1.606
10''
5.035
io“
14.683
10'
28.407
10^
32.678
10^
33.187
10^
33.239
;10*“
33.245
1337
Figure 6 on the other hand represents the low frequency torsional receptances
of the case in which the warping constraint is taken into consideration and
the ends are free to warp. This graph is included in order to show the variation
of fundamental torsional resonance frequencies for a range of Ktor values.
_ - K,3r=
K„,=
kK,„=
5*10':
1*10':
5-10"
1-10'
5-10'
1*10®
Figure 6. Fundamental Frequencies of Purely Torsional Vibrations
((3=0.01, x=0.13579[m], Warping Constraint Included, Ends are Free to Warp)
The second part of the study investigates the characteristics of doubly-coupled
vibrations. Now, the effects of each constraint are separately considered. A
channel having the given parameters is supported at its ends by springs Kt, Kr,
and Ktor- Warping constraint is included in the analysis and the ends are
assumed to warp freely. First analyzed is the effects of Ktor- For this Kt=10^°
[N/m] and Kr =10^° [N] are assigned at both ends of the channel and kept fixed
throughout the study. Ktor is varied and the frequencies are shown in Table 2.
Table 2. Effects of Ktnr in
Doublv-counled Vibrations
(Kt=10^° [N/m] and Kr=10^® [N], Warping Constraint Included)
A: First Torsion Dominated Frequency [Hz]
B: First Flexure Dominated Frequency [Hz]
Ktor [Nl
A
B
10-^
1.607
134.603
10''
5.069
134.936
10°
15.620
138.258
10'
39.725
169.247
10^
56.691
318.423
10^
59.528
422.061
10^
59.828
430.642
1338
0.0 0.2 0.4 0.6 0.8 1.0
ND LENGTH
Figure 2. Fundamental Mode Shapes of Purely Torsional Vibrations
((3=0, No Warping Constraint)
Then, the warping constraint To is included in the analysis and the results of
purely torsional vibrations are presented again. The beam had the same Ktor
values at both ends and the numerical values of the relevant parameters are
taken to be those previously defined. Figure 3 represents the fundamental
mode shapes for which the ends are free to warp, whereas Figure 4 shows the
mode shapes of the case in which there is no warping at the ends.
Figure 3. Fundamental Mode Shapes of Purely Torsional Vibrations
(P=0, Warping Constraint Included, Ends are Free to Warp)
(P=0, Warping Constraint Included, Ends Can Not Warp)
Figure 5 is drawn to highlight the effects of end warping. Both ends of the
channel are restrained with Ktor=l * lO’ [N] and all the other parameters of the
study are kept fixed. Figure 5 represents the direct torsional receptances of
two cases in which the ends of the channel are allowed to warp and not to
warp in turn. It can be seen that the prevention of end warping increases the
resonant frequencies.
Figure 5. Frequency Response of Purely Torsional Vibrations
([3=0.01, X =0.13579 [m], Ktor=l*10' [N]» Warping Constraint Included)
1340
It can be seen that, when it has lower values Ktor is more effective on the
torsion dominated resonance frequencies. For the higher Ktor values, the
effects are more apparent on the flexure dominated frequencies.
Figure 7 represents the direct flexural receptance of the channel for
a set of selected end stiffnesses. Torsion dominated resonances at 59.528 [Hz],
206.071 [Hz] and 476.649 [Hz] appear as spikes. The flexure dominated
resonance occurs at 422.061 Hz.
Figure 7. Frequency Response of Doubly-coupled Vibrations
(ti=0.001, P=0.001, x=0.13579[m], Warping Constraint Included,
Ends are free to warp, Kr=1.10^°[N], Kt=1.10^° [N/m], Ktor=l-10^ [N] )
Then the effects of the rotational spring, Kr, are considered. The ends of the
channel are assumed to be restrained with Kt=10^° [N/m] and Ktor =10^° [N].
The resulting frequencies are given in Table 3 for a range of Kr values.
Table 3. Effects of Kr in Doublv-coupled Vibrations
(Kt=10^° [N/m] and Ktor=10^° [N], Warping Constraint Included)
A; First Torsion Dominated Frequency [Hz]
B: First Flexure Dominated Frequency [Hz] _
Kr[N] _ A _ B
10*^
58.678
205.344
10°
58.679
205.358
10^
58.699
206.688
10^
58.858
218.055
10"
59.437
289.738
10^
59.799
405.667
10^
59.855
429.345
1341
It is seen that Kr is not effective on torsion dominated resonance frequencies,
but plays significant role for flexure dominated resonance frequencies.
Finally considered the effects of the transverse spring Kt. Again, the channel is
assumed to have very high Kr and Ktor values at both ends and Kt values are
varied. Table 4 shows the resonance frequencies.
Table 4. Effects of Kt in Doublv-coupled Vibrations
(Kr=10^° [N] and Ktor =10^° [N], Warping Constraint Included)
A: First Torsion Dominated Frequency [Hz]
B: First Flexure Dominated Frequency [Hz] _
Kt [N/m]
A
B
10'
1.369
70.455
10^
4.328
70.501
10^
13.592
70.973
10^
39.282
77.379
10=
58.095
159.665
10=
59.699
361.255
10*
59.860
431.215
It can be seen that the transverse stiffness, like torsional stiffness, effects both
flexure and torsion dominated frequencies.
4.CONCLUSIONS
In this study, a new analytical method is presented for the analysis of forced
vibrations of open section channels in which the flexible supports provide the
end constraints. The dynamic response of open section channels is a coupled
problem and their analysis requires the simultaneous consideration of all the
possible vibratory motions. The wave propagation approach is an efficient tool
for this complicated problem and the developed method is based on that.
The current method analyzes the forced, coupled vibrations of open section
channels. The channels, taken as Euler-Bernoulli beams, have uniform cross-
section and a single symmetry axis. That consecutively leads to the coupling
of flexural vibrations in one direction and torsional vibrations. The excitation
is assumed to be in the form of a harmonic point force, acting at the centroid.
1342
Various frequency response curves of uncoupled and coupled vibrations are
presented for a variety of different elastic end boundary conditions( which may
also have the inertial properties). The developed method, although aimed at
determining the forced vibration characteristics, is also capable of determining
the free vibration properties. This is also demonstrated by presenting various
mode shape graphs. It has been determined that the transverse and the torsional
stiffnesses play more significant role as compared to the rotational stifness.
The method can be used in analyzing the effects of multi point and/or
distributed loadings. This can simply be achieved by modifying the terms of
the forcing vector without increasing the order of the relevant matrix equation.
The developed method can also be used in the analysis of elastically
supported, triply-coupled vibrations of uniform channels. Results of that study
will be the subject of another paper.
REFERENCES
1. Gere, J.M. and Lin, Y.K., Coupled Vibrations of Thin-Walled Beams of
Open Cross-Section. J. Applied Mech Trans. ASME.,\9SZ, 80,373-8.
2. Lin, Y.K., Coupled Vibrations of Restrained Thin-Walled Beams.
J. Applied Mech. Trans. ASME., 1960, 82, 739-40.
3. Dokumaci, E., An Exact Solution for Coupled Bending and Torsional
Vibrations of Uniform Beams Having Single Cross-Sectional Symmetry.
JSoundandVib.Am, 119,443-9.
4. Bishop, R.E.D, Cannon, S.M. and Miao, S., On Coupled Bending and
Torsional Vibration of Uniform Beams. J.Sound and Fi'/)., 1989,131,457-64.
5. Cremer, L. and Heckl, y\..,Structure~ Borne Sound, Springer-Verlag,1988.
6. Mead, D.J. and Yaman, Y., The Harmonic Response of Uniform Beams on
Multiple Linear Supports: A Flexural Wave Analysis. J. Sound and Vib,
1990, 141,465-84
7. Yaman, Y. Wave Receptance Analysis of Vibrating Beams and Stiffened
Plates. PA Z). Ttew, University of Southampton, 1989.
8 Yaman, Y., Vibrations of Open-Section Channels: A Coupled Flexural and
Torsional Wave Analysis. (J. Sound and Vib, Accepted for publication)
1343
1344
THE RESPONSE OF TWO-DIMENSIONAL PERIODIC STRUCTURES
TO HARMONIC AND IMPULSIVE POINT LOADING
R.S. Langley
Department of Aeronautics and Astronautics
University of Southampton
Southampton S017 IBJ
ABSTRACT
Much previous work has appeared on the response of a two-dimensional
periodic structure to distributed loading, such as that arising from a harmonic
pressure wave. In contrast the present work is concerned with the response
of a periodic structure to localised forcing, and specifically the response of the
system to both harmonic and impulsive point loading is considered by
employing the method of stationary phase. It is shown that the response can
display a complex spatial pattern which could potentially be exploited to
reduce the level of vibration transmitted to sensitive equipment.
1. INTRODUCTION
Many types of engineering structure are of a repetitive or periodic
construction, in the sense that the basic design consists of a structural unit
which is repeated in a regular pattern, at least over certain regions of the
structure. An orthogonally stiffened plate or shell forms one example of an
ideal two-dimensional periodic structure in which the fundamental structural
unit is an edge stiffened panel. Although a completely periodic structure is
unlikely to occur in practice, much can be ascertained regarding the structural
dynamic properties of a real structure by considering the behaviour of a
suitable periodic idealization. For this reason, much previous work has been
performed on the dynamic behaviour of two-dimensional periodic structures,
with particular emphasis on free vibration and the response to pressure wave
excitation [1,2]. However, no results have yet been appeared regarding the
response of two-dimensional periodic structures to point loading (as might
arise from equipment mounts), and this topic forms the subject of the present
work. A general method of computing the response to both harmonic and
impulsive loading is presented, and this is then applied to an example system.
Initially the response of a two-dimensional periodic structure to harmonic
point loading is considered, and it is shown that the far-field response can be
expressed very simply in terms of the "phase constant" surfaces which
describe the propagation of plane waves. It is further shown that for
1345
excitation within a pass band two distinct forms of response can occur; in the
first case the amplitude of the response has a fairly smooth spatial distribution,
whereas in the second case a very uneven distribution is obtained and "shadow
zones" of very low response are obtained. The second form of behaviour is
related to the occurrence of caustics (defined in section 3.3), and the
distinctive nature of the response suggests that a periodic structure might be
designed to act as a spatial filter to isolate sensitive equipment from an
excitation source.
Attention is then turned to the impulse response of a two-dimensional periodic
structure. It is again shown that the response can be expressed in terms of the
phase constant surfaces which describe the propagation of plane waves. The
application of the method of stationary phase to this problem has a number of
interesting features, the most notable being the fact that four or more
stationary points can arise. It is found that a surface plot of the maximum
response amplitude against spatial position reveals features which resemble the
"caustic" distributions obtained under harmonic loading.
2. RESPONSE TO A HARMONIC POINT LOAD
2.1 Modal Formulation and Extension to the Infinite System
A two-dimensional periodic structure consists of a basic unit which is repeated
in two directions to form a regular pattern, as shown schematically in Figure
1. Each unit shown in this figure might represent for example an edge
stiffened curved panel in an aircraft fuselage structure, a three-dimensional
beam assembly in a roof truss structure, or a pair of strings in the form of a
"-f" in a cable net structure. The displacement w of the system can be
written in the form w{n,x), where n={n^ nf) identifies a particular unit and
x = {Xi X2 X3) identifies a particular point within the unit. The coordinate
system x is taken to be local to each unit, and the precise dimension of both
X and the response vector w will depend on the details of the system under
consideration.
The present section is concerned with the response of a two-
dimensional periodic structure to harmonic point loading of frequency co. In
the case of a system of finite dimension, the response at location (n,x) to a
harmonic force F applied at (0,jCo) can be expressed in the standard form [3]
»’(n,*)=EE
P 9
(1)
where rj is the loss factor, 4>p^(n,x) are the modes of vibration of the system
and (j}pg are the associated natural frequencies. The modes ^p^ which appear
in equation (1) are scaled to unit generalized mass, so that
1346
/!, «! V
(2)
where V represents the volume (or equivalent) of a unit and p(jc) is the mass
density. The present concern is with the response of an infinite system, or
equivalently the response of a large finite system in which the vibration decays
to a negligible level before reaching the system boundaries. In this case the
response is independent of the system boundary conditions, and it follows that
any analytically convenient set of modes can be employed in equation (1). As
explained in reference [4], it is expedient to consider the Born- Von Karman
(or "periodic”) boundary conditions, as in this case the modes of vibration can
be expressed very simply in terms of propagating plane wave components.
In this regard it can be noted from periodic structure theory [5] that a
propagating plane wave of frequency w has the general form
w(rt,jc)=i?^{^(jc)exp(zej«, +ie^n^nu)t)} , (3)
where and eo are known as the propagation constants of the wave (with -
T<e,<T and -7r<G2<T for uniqueness), and g{x) is a complex amplitude
function. By considering the dynamics of a single unit of the system and
applying Bloch’s Theorem [5], it is possible to derive a dispersion equation
which must be satisfied by the triad (w, 61,62) - by specifying Gj and €2 this
equation can be solved to yield the admissible propagation frequencies w. By
way of example, solutions yielded by this procedure for a plate which rests on
a grillage of simple supports are shown in Figure 2 (after reference [6]). It
is clear that the solutions form surfaces above the 61-62 plane - these surfaces
are usually referred to as "phase constant" surfaces, and a single surface will
be represented here by the equation a;= 0(61,62). The phase constant surfaces
always have cyclic symmetry of order two, so that 0(ei,62)=0(-6i,-62); for an
orthotropic system the surfaces also have cyclic symmetry of order four, and
therefore only the first quadrant of the 61-62 plane need be considered
explicitly, as in Figure 2.
The key point about the Born- Von Karman boundary conditions is that
a single propagating wave can fully satisfy these conditions providing and
6o are chosen appropriately. The conditions state that the left hand edge of the
system is contiguous with the right hand edge, and similarly the top edge is
contiguous with the bottom edge, so that the system behaves as if it were
topologically equivalent to a torus. If the system is comprised of XN2
units, then a propagating wave will satisfy these conditions if and
62^2 =2x^ for any integers p and q. Following equation (3), the displacement
associated with such a wave can be written in the form
1347
(4)
where ei^ and €2, are the appropriate values of the phase constants, and
<^^^=0(ej^,e2g)- Now since it follows that a wave of
frequency travelling in the opposite direction to will also satisfy the
boundary conditions. This wave say) will have the form
where it has been noted from periodic structure theory that reversing the
direction of a wave leads to the conjugate of the complex amplitude function
^(x). The two waves represented by equations (4) and (5) can be combined
with the appropriate phase to produce two modes of vibration of the system
in the form
KSn,x)
•
Re
Im
(6)
By adopting this set of modes it can be shown [4] that equation (1) can be re¬
expressed as
K.(«,x)= £ £ 2g;WF^g„(xJexp(-ie,^«,-%,n,)
^=1-^/2 9=1 -Nj/2 +Z?7) -(J?
where and M have been taken to be even, and the amplitude function gp^
is scaled so that
g^,=[2p(x)WV,Ar,]-%(x), {llV)\^f„(x)f;,(x)dx=\, (8,9)
where the normalized amplitude function fp^ is defined accordingly. The
summation which appears in equation (7) includes only those modes associated
with a single phase constant surface 0(€i,62); if more than one surface occurs
then the equation should be summed over the complete set of surfaces. The
summation will include modes for each surface, which is consistent with
known results for the modal density of a two-dimensional periodic structure.
Equation (7) yields the response of a finite system of dimension XM
to a harmonic point load - this response is identical to that of an infinite
system if the vibration decays to a negligible amount before meeting the
system boundaries. If the system size is allowed to tend to infinity in equation
1348
(7) then neighbouring values of the phase constants e^p and €2^ become closely
spaced (since deip=ei_p+i-eip=2T/A^i and de2g= €2,9+ 2x77/2) > and in this
case the summations can be replaced by integrals over the phase constants to
yield
- LI me„e,)ninv)-c^^
where 0)= 0(61,62) and ^(x) is the complex amplitude associated with the wave
(£0,61,62). The evaluation of the integrals which appear in equation (10) is
discussed in the following sub-sections.
2.2 Integration over 61
The integral over 6i which appears in equation (10) can be evaluated by using
contour integration techniques. Two possible contours in the complex e, plane
are shown in Figure 3; to ensure a zero contribution from the segment
Im(6i) = ±oo, the upper contour is appropriate for /Zi<0 while the lower
contour should be used for n^X). For each contour the contributions from
the segments and 61;;= x cancel, since the integrand which appears in
equation (10) is unchanged by an increment of 2x in the real part of ei. The
only non-zero contribution to the integral around either contour therefore
arises from the segment which lies along the real axis. The poles of the
integrand occur at the 61 solutions of the equation
[fl(6i,62)?(U/i7)-a;^=0, (11)
for specified 62 and oj. By definition there will be two real solutions^ in the
absence of damping (77=0) providing the frequency range covered by the
phase constant surface includes oj. Any complex solutions to equation (11) in
the absence of damping will correspond physically to "evanescent" waves
which decay rapidly away from the applied load. The present analysis is
concerned primarily with the response of the system in the far field (that is,
at points remote from the excitation source), and for this reason attention is
focused solely on those roots to equation (11) which are real when 77=0. The
effect of damping on these roots can readily be deduced: if 77 is small then it
follows from equation (11) that a real solution 6^ will be modified to become
6I-i(o7/2)(5Q/^6l)■^ and hence the real pole for which dn/36i<0 is moved to
the upper half plane, while that for which 30/56, >0 is moved to the lower
half plane. Given that the residue at such a pole is proportional to (30/36i)'\
^One positive and one negative. These solutions will have the form ±6,
for an orthotropic system.
1349
it follows that the sign of the residue which arises from the contour integral
is determined by the integration path selected, and hence by the sign of
These considerations lead to the result
~Tr~l fi|3fi/3«,|(l+/)/)
(12)
where e,(£2,M) is the appropriate solution to equation (11). The evaluation of
the integral over €3 is discussed in the following section.
3.3 Integration over €3
Since the present concern is with the response of the system at some distance
from the excitation point, the integral over e, which appears in equation (12)
can be evaluated to an acceptable degree of accuracy by using the method of
steepest descent [7], With this approach it is first necessary to identify the
value of €2 for which the exponent -i(eirti+e2«2) is stationary. The condition
for this is
{3ejde^n^+n^=0. (13)
Now Gi and satisfy the dispersion relation, equation (11), and thus equation
(13) can be re-expressed in the form
(aQ/a62)«r(5^2/a€,)«2=o, (i4)
where it has been noted from equation (11) that, for fixed co, 3ei/3€2=-
(5Q/3e2)/(9fi/36i), In the absence of damping the wave group velocity lies in
the direction (SQ/Sei and in this case it follows from equation (14)
that the group velocity associated with the required value of €2 is along («i ru).
For light damping this result will be substantially unaltered, although damping
will have an important effect on the value of the exponent -\{€^n^-\-e2iv^ at the
stationary point. This effect can be investigated by noting initially that
d{e^n^+&^n^)IBr}~{deJbri B&Jbr]).{n^ n^. (15)
Now it follows from equation (11) that for light damping {ri<l)
(dQ/de, dQlde,).{de,ldr] de^/dv) = -io)l2, (16)
and hence equations (14)-(16) can be combined to yield the following result
at the stationary point
d{e^n^+e^n2)/dr) = -io)n/2c^. (17)
1350
Here Cg=^{{d£l!dex?+{d^lbe^'^] is the resultant group velocity and
n=V[ni+ni-'\ is the radial distance (in units) from the excitation point to the
unit under consideration. It follows that in the immediate vicinity of the
stationary point the exponent can be expanded in the form
(18)
where the subscript 0 indicates that the term is to be evaluated at the
stationary point under the condition 7]=0; for ease of notation, this subscript
is omitted in the following analysis. The method of steepest descent proceeds
by substituting equation (18) into equation (12) and assuming that: (i) the main
contribution to the integral arises from values of 62 in the immediate vicinity
of the stationary point; (ii) the integrand is effectively constant in this vicinity,
other than through variation of the term e2-(e2)o which appears in equation
(18); (iii) under conditions (i) and (ii) the integration range can be extended
to an infinite path without significantly altering the result. The method then
yields [7]
w{n,x) = ~if *F7o[20V|aQ/a£j/2xp(A:)p(A:o)|«i(aV5e2)| .
(19)
where/ is the normalized complex wave amplitude which appears in equation
(9), and all terms are to be evaluated at the stationary point.
The stationary point associated with equation (19) is that point for
which the group velocity is in the {n^ rQ direction. Geometrically, this is the
point at which the normal to the curve a)=Q(ei, £2) itt the plane lies in the
{n^ Ho) direction. Three such curves are shown schematically in Figure 4,
together with a specified (n, 722) direction. For the frequencies and coj the
situation is straight forward, in the sense that a unique stationary point exists
for any {n^ n^) direction. For the frequency 0J2 the situation is more complex,
since: (i) two stationary points occur for the (n^ rQ direction shown, and (ii)
no stationary point exists if the {n^ direction lies beyond the heading B
shown in the figure (the dashed arrow represents the normal with maximum
inclination to the axis). In case (i) equation (19) should be summed over
the two stationary points, while in case (ii) the method of steepest descent
predicts that w{n,x) will be approximately zero, leading to a region of very
low vibrational response. If the direction («i coincides with the dashed
arrow, then equation (19) breaks down, since it can be shown that
at this point. The heading indicated by the dashed arrow represents a caustic
[7], and the theory given in the present section must be modified for headings
1351
(Wi Wj) which are in the immediate vicinity of the caustic - full details of the
appropriate modifications are given in reference [4]. An example which
illustrates the application of equation (19) is given in section 4.
3. RESPONSE TO AN IMPULSIVE POINT LOAD
If the system is subjected to an impulsive (i.e. a delta function applied at
r=0), rather than harmonic, point load, then equation (10) becomes [8]
w{n,x,t)={N^NJ2Tp) f [ g*
II (20)
where co = ^](€i,€2)- The method of stationary phase can be applied to this
expression to yield [8]
w{n,x,t)-{\l2yr)[p{x)p{x)\J\r‘^^^^
exp(-/Gjtti -k^n^nQt+ib) } ,
(21)
where all terms are evaluated at the stationary point, and J and 5 are defined
as
/=(a"n/a6?)(a"Q/ae^-(3"0/ae,a6,)", S=(Tr/4)sgn(a%/fle?){l+sgn(/)}.
(22,23)
In this case the stationary point is given by the solution to the equations
=(afi =(30 /3e,>. (24,25)
In practice equations (24) and (25) may yield multiple solutions (stationary
points), in which case equation (21) should be summed over all such points.
Furthermore, stationary points having 7=0 indicate the occurrence of a
caustic, and equation (21) must be modified in the immediate vicinity of such
points as detailed in reference [8]. An example of the application of equation
(21) is given in the following section.
1352
4. EXAMPLE APPLICATION
4. 1 The System Considered
The foregoing analysis is applied in this section to a two-dimensional periodic
structure which consists of a rectangular grid of lumped masses m which are
coupled through horizontal and vertical shear springs of stiffness and h
respectively. Each mass has a single degree of freedom consisting of the out-
of-plane displacement w, and a linear spring of stiffness k is attached between
each mass and a fixed base. It is readily shown that the system has a single
phase constant surface of the form
Q -(ep€,)=iLii(l-cos€i)+/x2(l-cos€2)+a;^,
where ix^^lkjm, and o)^-==klm. The function U can be used in
conjunction with the analysis of the previous sections to yield the response of
the system to harmonic and impulsive point loading; in this regard it can be
noted that for the present case p{x)=m, V—l, and /(:»:) = 1.
4.2 Response to Harmonic Loading
The surface is shown as a contour plot in Figure 5 for the case
m=1.0, oj„==0, Ati = 1.0, Results for the forced harmonic response
of this system at the two frequencies w = 1.003 and cx> = 1.181 are shown in
Figures 6 and 7. In each case the response of a 40 x40 array of point masses
is shown; a unit harmonic point load is applied to mass (21,21) and the loss
factor is taken to be =0.05. Two sets of contours are shown in each Figure:
the smooth contours have been calculated by using equation (19) while the
more irregular contours have been obtained by a direct solution of the
equations of motion of the finite 1600 degree-of-freedom system. By
considering the results shown in Figure 6, it can be concluded that: (i) for the
present level of damping the finite system effectively behaves like an infinite
system, and (ii) the analytical result yielded by equation (19) provides a very
good quantitative estimate of the far field response. It can be noted from
Figure 5 that no caustic occurs for a) = 1.003, in the sense that equation (14)
yields only one stationary point which contributes to equation (19). In
contrast, a caustic does occur for the case a) = 1.181, and this leads to the very
irregular spatial distribution of response shown in Figure 7. Two stationary
points contribute to equation (19), and constructive and destructive
interference between these contributions is responsible for the rapid
fluctuations in the response amplitude. It is clear that the response exhibits
a "dead zone" for points which lie beyond the caustic heading (in this case
30.25° to the «i-axis), as predicted by the analysis presented in section 2.
4.3 Response to Impulsive Loading
The impulse response of a system having m=1.0, /xi = 1.0, ^2=0. 51, and
1353
co„'“0.25 has been computed. The impulse was taken to act at the location
/Zi=«2=0 and the time history of the motion of each mass in the region -
iO<(72i,n2) ^ 10 was found by using equations (21)-(23). For each mass the
maximum response \w\ was recorded, and the results obtained are shown as
a contour plot in Figure 8. In accordance with Fourier’s Theorem, the
impulse response of the system contains contributions from all frequencies,
and therefore the spatial distribution of | w| can be expected to lie somewhere
between the two extreme forms of harmonic response exhibited in Figures 6
and 7. This is in fact the case, and the response shown in Figure 8 retains a
distinctive spatial pattern. As discussed in reference [8], the results shown in
Figure 8 are in good agreement with direct simulation of the impulse response
of the system.
5. CONCLUSIONS
This paper has considered the response of a two-dimensional periodic structure
to both harmonic and impulsive point loading. With regard to harmonic
loading, it has been shown that the spatial pattern of the response is strongly
dependent on the occurrence of a caustic: if no caustic occurs then the
response has a fairly smooth spatial distribution, whereas the presence of a
caustic leads to an irregular spatial distribution and a "dead zone" of very low
response. This type of feature is also exhibited, although to a lesser degree,
in the spatial distribution of the response to an impulsive point load. This
behaviour could possibly be exploited to reduce vibration transmission along
a specified path, although the practicality of this approach for a complex
system has yet to be investigated. The present analytical approach can be
applied to all types of two-dimensional periodic structure - the information
required consists of the phase constant surface(s) 0(61,62) and the associated
wave form(s) f{x), both of which are yielded by standard techniques for the
analysis of free wave motion in periodic structures [1,2].
REFERENCES
1. S.S. MESTER and H. BENAROYA 1995 Shock and Vibration 2, 69-
95. Periodic and near-periodic structures.
2. D.J. MEAD 1996 Journal of Sound and Vibration 190, 495-524.
Wave propagation in continuous periodic structures: research
contributions from Southampton 1964-1995.
3. L. MEIROVITCH 1986 Elements of Vibration Analysis, Second
Edition. New York: McGraw-Hill Book Company.
4. R.S. LANGLEY 1996 Journal of Sound and Vibration (to appear).
The response of two-dimensional periodic structures to point harmonic
forcing.
1354
1
■
■
■
1
■
■
■
0
m
■
1
■
Xj. ■j'
Oi
< -
^1
1 ^
Figure I , Schematic of a two-dimensional periodic structure. The arrow indicates' the reference unit
(with n=0) while the circle represents a general point (re,x). The structure may have a third spatial
coordinate X3, which for convenience is not shown in the figure. The point load considered in section
3 is applied at the location of the arrow.
Figure 2. Phase constant surfaces for a plate which rests on a square grillage of simple supports.
Q is a non-dimensional frequency which is def as Q=a)LV(m/D), where m and D are respectively
the mass per unit area and the flexural rigidii^ the plate, and L is the support spacing.
1355
Figure 5. Contour plot of the phase constant surface for the case ^=0.57. The contours
are separated by an increment Aco =0. 1477. The two contours considered in section 4.2 are indicated
as follows: (a) a) = 1.033; (b) a) = 1.181.
Figure 6. Response |H'(n,x)|2 of the 40x4v .uass/spring system to a unit harmonic force of
frequency w- 1.033 applied at the location i=y=21. The contours correspond to the response levels
lK«,x)!'=0.01, 0.02, andO.05.
1357
STICK-SLIP MOTION OF AN ELASTIC SLIDER SYSTEM ON
A VIBRATING DISC
HOuyang J E Mottershead M P CartmeU ' MIFiiswell^
Department of Mechanical Engineering, University of Liverpool
^ Department of Mechanical Engineering, University of Edinburgh
^ Department of Mechanical Engineering, University of Wales Swansea
ABSTRACT
The in-plane vibration of a slider-mass which is driven around the surface of a
flexible disc, and the transverse vibration of the disc, are investigated. The disc
is taken to be an elastic annular plate and the slider has flexibility in the
circumferential (in-plane) and transverse directions. The static fiiction
coefi&cient is assumed to be higher than the kinetic friction. As a result of the
fiiction force acting between the disc and the slider system, the slider will
oscillate in the stick-slip mode in the plane of the disc. The transverse vibration
induced by the slider will change the normal force of the slider system acting
on the disc, which in turn will change the in-plane oscillation of the slider. For
different values of system parameters, the coupled in-plane oscillation of the
slider and transverse vibration of the disc will exhibit quasi-periodic as well as
chaotic behaviour. Rich patterns of chaotic vibration of the slider system are
presented in graphs to illustrate the special behaviour of this non-smooth
nonlinear dynamical system The motivation of this work is to analyse and
understand the instability and/or squeal of physical systems such as car brake
discs where there are vibrations induced by non-smooth dry-fiiction forces.
NOMENCLATURE
a , b mner and outer radii of the annular disc
c damping coefficient of the slider in in-plane direction
h thickness of the disc
i = V=T
1359
k , k transverse and in-plane stiffiiess of the slider system
m
r
t
^stiek
mass of the slider
radial co-ordinate in cylindrical co-ordinate system
radial position of the slider
modal co-ordinate for k nodal circles and / nodal diameters
time
the time of the onset of sticking
u , Mp transverse displacement of the slider mass and its initial value
w , Wq transverse displacement of the disc and its initial value
flexural rigidity of the disc
Kelvin-type damping coefficient
Young’s modulus
initial normal load on the disc jfrom the slider system
total normal force on the disc from the slider system
combination of Bessel functions representing mode shape in radial
direction
circumferential co-ordinate of cylindrical co-ordinate system
kinetic and static dry fiction coefficient between the shder and the
D
D'
E
N
P
R.
0
disc
V
P
(P
^ stick
¥
¥ki
CO.
Poisson’s ratio
damping ratio of the disc
specific density of the disc
absolute circumferential position of the slider
absolute circumferential position of the slider when it sticks to the disc
circumferential position of the slider relative to the drive point
mode function for the transverse vibration of the disc corresponding to
^kl
natural (circular) frequency correq)onding to <5^^
1360
Q constant rotating speed of the drive point around the disc in radians per
second
INTRODUCTION
There exists a whole class of mechanical systems which involve discs rotating
relative to stationery parts, such as car brake discs, clutches, saws, computer
discs and so on. In these systems, dry-friction induced vibration plays a crucial
role in system performance. If the vibration becomes excessive, the system
might fail, or cease to perform properly, or make offensive noises. In this
paper, we investigate the vibration of an m-plane slider system, with a
transverse mass- spring- damper, attached through an in-plane spring to a drive
point which rotates at constant speed around an elastic disc, and the vibrations
of the disc. Dry friction acts between the sHder system and the disc.
Dry-friction induced vibration has been studied extensively [1-4]. For car brake
vibration and squeal, see the review papers [5,6]. The stick-slip phenomenon of
dry-friction induced vibration is studied in the context of chaotic vibration [7-
10]. Popp and Stelter [7] studied such motion of one and two degrees of
freedom system and foimd chaos and bifiircation. They also conducted
experiments on a beam and a circular plate (infinite number of degrees of
freedom). These theoretical works are about systems of less than three degrees
of freedom, and the carrier which activates the friction is assumed to be rigid.
In this paper, we consider an elastic disc so that the transverse vibrations of the
disc are important. As a result of including the transverse vibrations of the disc,
rich patterns of chaos, which have not been reported previously are found. If
there is only shding present at constant speed, the problem is reduced to a
linear parametric analysis which was carried out for a pin-on- disc system in
[11] and for a pad-on-disc system in [12,13].
m-PLANE OSCILLATION OF THE SLIDER SYSTEM
As the drive point, which is connected to the shder-mass through an in-plane,
elastic spring, is rotated at constant angular speed around the disc, the driven
slider will undergo stick-slip oscillations. The whole system of the shder and
the disc is shown in Figure 1.
The equation of the in-plane motion of the slider system relative to the rotating
drive point, in the sliding phase, is.
1361
(1)
while in sticking, the equation of the motion becomes.
The relationship between the relative motion of the slider system to the drive
point and its absolute motion (relative to the stationary disc) is
(p = Qt + y/, (3)
We consider the foEowing initial conditions which are intended to simulate
what happens in a disc brake. The slider system is at rest and there is no normal
loading on the disc j&om the slider. Then a constant normal load is applied
which causes transverse vibrations in the disc. At the same time, the drive point
is given a constant angular velocity. Other initial conditions are possible, so
that there is no loss of generality.
First, sliding from the initial sticking phase occurs when,
The slider will stick to the disc agaiu when,
ju^P (during sliding), (5)
or it will begin to sHde again iJ^
ij/ = , \k^rQtp-\ > (dxiring sticking). (6)
Consequently, the slider system will stick and slide consecutively on the disc
surface.
TRANSVERSE VIBRATION OF THE ANNULAR DISC
The equation of motion of the disc under the slider system is,
ph^ + D'V^'w + DW = --5(r-r„)5i$-<p)P. (7)
a/" r
The total force P is the summation of initial normal load N and the resultant
of the transverse motion w of the slider. Its expression is,
P - N + mu + cu-h kiu-u^) . (8)
Since it is assumed that the slider system is always in perfect contact with the
disc, then,
1362
(9)
u{t) = w{r„(p{t)A-
Substitution of ecpiations (8) and (9) into (7) leads to.
ph
^ — + /:)VV = --5(r - r, )6(^ - (p)[N +
dt
dt
^..dw d^w d^w .dw dw
(10)
k(w-wj\.
Note that equation (10) is valid whether the slider system is sticking or sliding.
When the slider sticks to the disc, equation (10) reduces to.
+ D-V‘ ^ +£)W = --5(r - )5(5 -<p)x
St St r (11)
d'^W ^ M
[A^+m— + c— + A:(w-w„)].
COUPLED VIBRATIONS OF THE SLIDER AND THE DISC
Assume that the transverse motion of the disc can be represented by,
M>{r,e,t)=ttii/,{r,0)q,it), (12)
Jt=0 /=-<io
and,
where {r) is a combination of Bessel functions satisfying the boundary
conditions in radial direction at the inner radius and outer radius of the disc.
The modal functions satisfy the ortho-normality conditions ofj
(14)
1363
Equations (10) and (11) can be simplified by being written in terms of the
modal co-ordinates from equation (12).
During sticking, the motion of the whole system of the slider and the disc can
be represented by,
N
+ 2^0) -
— Z^„('-o)^«('-o)exp[i(5-0(»]x (15)
ph.U ''=0 s=-«o
{mq„+cq^+k{q^-q„{^)\).
The sticking phase can be maintained i^
rA\¥\</^XJ^+-
^Z Z ^«(?')exp(i/?»)x
{mq„+cq„+ k[q„ - (0)] } ] .
While in sliding, the motion of the whole system can be represented by.
%+2^a>^,qu+a>lq„ =
^^«('i)exp(-i/<»)-
-rrri Z^„('-o)^«(n,)exp[i(5-0«!’]x
pnO r=<i 5^=-oo
H?,. +i2sw„ +isw„) +
k{q„-qS'^)]},
r„{m{j/ + kw) = -/i,Sign((Z))[A'' +-
^Z Z KMx
exp(LS(Z>)M?„ +i2s^„ +(ls^-5>^)9„] +
c{q„ +>s(?g„) +*[!?„ -?„(o)]}].
The sliding phase can be maintained if,
1364
\K^M< mXN +-J^t t K{r,)es.p{is(p)x
■yjphb ^=0^^
{iTiq„+i2sj>q„+(is^-s^j)^)qJ+ (20)
c(?„ +isj^J + k[q„ -9„(0)]}],
when,
\j/ = ~0 or ^ = 0. (21)
COMPUTING PROCESS
As the shder system sticks and sHdes consecutively, the governing equations of
the coupled motions of the whole system switch repeatedly from equations
(15), (16) and (17) to equation (18), (19) and (20). TTie system is not smooth.
Since the condition which controls the phases of the slider system itself
depends on the motions, it is also a nonlinear system, whether is a constant
or a function of relative speed ^ . In order to get modal co-ordinates, we have
to truncate the mfinite series in equation (12) to jfinite terms. Then numerical
integration is used to solve equations (15), (18) and (19). Here a fourth order
Rimge-Kutta method is used for second order simultaneous ordinary
differential equations.
Since equation (18) has time-dependent coefficients, time step length has to be
very small. Constant time step lengths are chosen when the m-plane slider
motion is well within the sticking phase or the shding phase in the numerical
integration. As it is imperative that the time step should be chosen such that at
the end of some time intervals the shder happens to be on the sticking- shding
interfaces, we use a prediction criterion to choose next time step length when
approaching these interfaces. Therefore, at the sticking-shding interfaces, the
time step length is variable (actuahy smaher than it is while weh within sticking
or shding). Nevertheless, tbe interfaces equations ( equation (17) or equations
(20-21) ) are only approximately satisfied [10].
When transverse motion of the shder system becomes so violent that the total
normal force P in it becomes negative or becomes several times larger than
the initial normal load 7/ , we describe the system as being unstable. Then the
motion begins to diverge. But this instabihty should be distinguished from a
chaotic motion which is bounded but never converge to a point.
1365
NUMERICAL EXAMPLES
The following data are used in the computation of numerical examples:
a = 0.065m, b = 012m, OTm, /z = 0.001m; = 120GPa, v= 0.35,
Z)*= 0.00004; yW, =0.4, //^=0.24, A: = lOOON/m, = lOON/m,
m = 0.1kg, p = 7000kg / m^ The disc is clamped at inner radius and free at
outer radius. Note that in these numerical examples, the disc thickness is
dehherately taken to be very small in order to reduce the amount of computing
work. However, this will not affect the qualitative features of the results or
conclusion drawn from the results thus obtained. The first five natural
(circular) frequencies are 451.29, 462.73, 426.73, 508.23, 508.23. We will
concentrate on the vibration solutions at different levels of initial normal load.
But occasionally solutions at different rotating speed or different damping
ratios are investigated. Unless specified expressly, the Poincare sections are for
the in-plane vibration of the slider system
First of all, we study the effect of the normal loadA^. Take f2=l0 and
^ — When N is very small^ the Poincare section is a perfect ellipse
which indicate the in-plane vibration of the slider system is quasi-periodic, as
the transverse vibration of the disc is too small to affect total normal force P .
A typical plot of such motion is shown in Figure 2 for A^=0.5kPa. As N
increases, the sticking period gets longer, the bottom part of the elhpse evolves
into a straight line, indicating phase points within the sticking phase. One of
such plots is given in Figure 3 for N =3kPa. A further increase of N not only
lengthens the straight line part of the Poincare section, but also creates an
increasingly ragged outline in the arch part of the plot. The curve is no longer
smooth and it seems that the in-plane motion begins to enter a chaotic state
from the quasi-periodic state. Figure 4 presents the Poincare section plot for
N =7.5kPa. There is a transition period from quasi-periodic motion to chaotic
motion, extending from N =6kPa up to N =9kPa. Chaos becomes detectable
at iV=10kPa, whose Poincare section is shown in Figure 5. Then chaotic
vibration follows. When N ~15kPa, the arch part of the Poincare section
becomes so fuzzy and thick that it should no longer be considered as a curve,
but rather a narrow (fractal) area. A hlow-up’ view of the arch part reveals
that phase points are distributed across the arch. Both plots are ^own in
Figure 6. Between iV'=17.ikPa and 18.325kPa, the vibration of the slider
enters a new stage, with Poincare sections looking like star clusters as
illustrated in Figures 7 and 8. This kind of motions are rather extraordinary and
1366
have not been reported in other works on stick-shp motions with a rigid
carrier. Afterwards, the ‘arch-door’ hke Poincare sections come back (see
Figure 9). The difference from previous Poincare sections of lower N is that
the new Poincare sections look like overlapping of earlier Poincare sections,
which indicates a clear layered structure, as diown in Figure 10 and more
obviously in the left hand side of Figure 11. At this stage, the vibration is very
chaotic. To give the reader a better picture, the Poincare section of a fixed
point on the disc at ( = 0.1m and = 0 ), is also shown in the right hand side
of Figure 11. The Poincare sections of the slider-mass and a point on the disc
are also given in Figures 12-15. In Figure 12 for A^=30.5kPa, the vibration
goes unstable. Here again, the Poincare sections have not been reported
elsewhere.
If disc damping is increased, vibration will become more regular, as shown in
Figure 13. Comparing Figures 11 with 13, we see that increase of disc damping
makes the vibration more concentrated though not always smaller. Unstable
vibration can be stabilised with more disc damping, as seen from Figure 14.
If there is no damping at aU, the resulting vibration due to dry fiiction will be
unstable, even at very small normal load N . In Figure 15, the motion of the
slider tends to run away in the tangential direction from the normal ellipse
attractor, while the motion of the disc goes unbounded.
Increasing the speed of the drive point seems to make vibration more chaotic
and more unstable, as shown in Figures 16-18. At this stage, however, we are
unable to make a definite conclusion on rotating speed as there might be
intervals of regular motions and intervals of chaotic motion for . More
numerical examples must be computed to draw a positive conclusion on this
parameter.
The correlation dimension is not a good measure of the vibration for the
current problem because its values fluctuate in some numerical examples. This
failure was perhaps first discovered in [7]. The reason can be either that the
system is non-smooth, or that the system has multiple degrees of freedom, or
both. Therefore, the correlation dimension or any other fractal dimensions is
not presented in this paper.
CONCLUSIONS
In this paper, we studied the in-plane stick-slip vibration of a slider system with
a transverse mass-spring-damper driven around an elastic disc through a spring
from a constant speed drive point, and transverse vibrations of the disc. The
1367
whole system had been reduced to six degrees of freedom after simplification.
From numerical examples computed so far, we can conclude that:
1. Both vibrations are very complex as this is a multi-degree of freedom, non¬
smooth system Rich patterns of chaotic vibration are found. Some have not
been reported elsewhere.
2. For the normal pressure parameter, smaller values allow quasi-peiiodic
solutions. Greater pressures result in chaotic motions. At certain large
pressures, the vibrations become unstable.
3. Disc dartping makes vibration more concentrated to smaller areas and when
sufficiently large it can stabilise otherwise unstable vibration.
4. An increase in the rotating speed can make the vibration more chaotic or
more unstable.
5. Correlation dimension is not a good measure of the vibration of this multi¬
degree of freedom, non-smooth dynamical system
6. Much more investigation needs to be earned out in understanding and
characterising the vibration of multi- degree of freedom, non- smooth dynamical
systems.
ACKNOWLEDGEMENT
This research is supported by the Engineering and Physical Sciences Research
Council (grant niunber J35177) and BBA Friction Ltd.
REFERENCES
1. Nakai, M., Chiba, Y. and Yokoi, M., Railway wheel squeal. Bulletin of
JSME, 1984, 27, 301-8
2. Lin, Y-Q and Wang Y-H, Stick-sHp vibration of drill strings. XEng.Ind.,
TransASME, 1991, 113, 38-43
3. Ferri, A. A. and Bindemann, A.C., Damping and vibration of beams with
various types of fiictional support conditions. J. VibAcoust, TransASME^
1992, 114, 289-96
4. Lee, A.C., Study of disc brake noise using multi-body mechanism with
friction interface. In Friction-Induced Vibration, Chatter, Squeal, and
Chaos, Ed. Ibrahim, KA. and Soom, A., DE-Vol. 49, ASME 1992, pp.99-
105
1368
5. Ibrahim, R.A., Friction-induced vibration, chatter, squeal, and chaos. In
Friction-Induced Vibration, Chatter, Squeal, and Chaos, Ed. Ibrahim,
KA. and Soom, A, DE-VoL 49, ASME 1992, pp. 107-38
6. Yang, S. and Gibson, R.F., Brake vibration and noise: reviews ,comments,
and proposed considerations. Proceedings of the 14th Modal Analysis
Conference, The Society of Experimental Mechanics, Inc., 1996, pp.l342-
9
7. Popp, K and Stelter, P., Stick-slip vibration and chaos. Phil. Trans. R Soc.
Lond. A(1990), 332, 89-106
8. Pfeiffer, F. and Majek, M., Stick-slip motion of turbine blade dampers. Phil.
Trans. R Soc. Lond. A(1992), 338, 503-18
9. Wojewoda, J., Kapitaniak, T., Barron, R. and Brindley, J., Complex
behaviour of a quasiperiodically forced experimental system with dry
friction. Chaos, Solitons and Fractals, 1993, 3, 35-46
10. Wiercigroch, M., A note on the switch function for the stick-slip
phenomenon. J.SoundVib., 1994, 175, 700-4
11. Chan, S.N., Mottershead, J.E. and Cartmell, M.P., Parametric resonances
at subcritical speeds in discs with rotating frictional loads. Proc. Instn.
Mech. Engrs, 1994, 208, 417-25
12. Mottershead, J.E., Ouyang, R, Cartmell, M.P. and Friswell, M.I.,
Parametric Resonances in an annular disc, with a rotating system of
distributed mass and elasticity; and the effects of friction and dan:q)ing.
Proc. Royal Soc. Lond. A., 1997, 453, 1-19
13. Ouyang, H., Mottershead, J.E., Friswell, M.l. and Cartmell, M.P., On the
prediction of squeal in automotive brakes. Proceedings of the 14th Modal
Analysis Conference, The Society of Experimental Mechanics, Inc., 1996,
pp. 1009-16
Figure 1. Slider system and disc in cylindrical co-ordinate system
1369
Figure 9. iV=21kPa
Figure 10. iV=24kPa
1374
A Finite Element Time Domain
Multi-Mode Method For Large Amplitude
Free Vibration of Composite Plates
Raymond Y. Y. Lee, Yucheng Shi and Chuh Mei
Department of Aerospace Engineering
Old Dominion University, Norfolk, VA 23529-0247
Abstract
This paper presents a time-domain modal formulation using the finite element method for
large-amplitude firee vibrations of generally laminated thin composite rectangular plates. Accurate
fi'equency ratios for fundamental as well as higher modes of composite plates at various maximum
deflections can be determined. The selection of the proper initial conditions for periodic plate
motions is presented. Isotropic beam and plate can be treated as special cases of the composite
plate. Percentage of participation from each linear mode to the total plate deflection can be
obtained, and thus an accurate frequency ratio using a minimum number of linear modes can be
assured. Another advantage of the present finite element method is that the procedure for obtaining
the modal equations of the general DujB5ng-type is simple when compared with the classical
continuum Galerkin’s approach. Accurate frequency ratios for isotropic beams and plates, and
composite plates at various amplitudes are presented.
Introduction
Large amplitude vibrations of beams and plates have interested many
investigators [1] ever since the first approximate solutions for simply supported
beams by Woinowsky-Kiieger [2] and for rectangular plates by Chu and Herrmann
[3] were presented. Singh et al. [4] gave an excellent review of various formulation
and assumptions , including the finite element method for large amplitude firee
vibration of beams. Srirangaraja [5] recently presented two alternative solutions,
based on the method of multiple scales (MMS) and the ultraspherical polynomial
approximation (UP A) method, for the large amplitude firee vibration of a simply
supported beam. The fi'equency ratios for the fundamental mode, ca/C0L> at the ratio
of maximum beam deflection to radius of gyration of 5.0 (Wmax/r =5.0) are 3.3438
and 3.0914, using the MMS and the UPA method, respectively. Eleven firequency
ratios including nine firom reference [4] were also given (see Table 1 of reference
[5]). It is rather surprising that the firequency ratio for the fundamental mode at
Wn«x/r =5.0 for a simply supported beam varied in a such wide range: fi-om the
lowest of 2.0310 to the highest of 3.3438, and with the elliptic function solution by
Woinowsky-Kiieger [2] giving 2.3501. Similar wide spread exists for the vibration
of plates. Rao et al [6] presented a finite element method for the large amplitude
firee flexural vibration of unstiffened plates. For the simply supported square plate.
1375
frequency ratios from six different approaches were reported (see Table 1 of
reference [6]). The frequency ratio at Wmax/h =1.0 varied from a low of 1.2967 to a
high of 1.5314, with Chu and Herrmann’s analytical solution [3] at 1.4023.
This paper presents a finite element time domain modal formulation for the large
amplitude free vibration of composite plates. The formulation is an extension from
the isotropic plates [7], and the determination of initial conditions for periodic
motions was not employed in reference [7]. The convergence of the fundamental
frequency ratio is investigated for a simply supported beam and a simply supported
square plate with a varying number of finite elements and a varying number of
linear modes. Accurate frequency ratios for fundamental and higher modes at
various maximum deflections, and percentages of participation from various linear
modes, are obtained for beams and composite plates.
Formulation
Strain-Displacement and Constitutive Relations
The von Karman strain-displacement relations are applied. The strains at any
point z through the thickness are the sum of membrane and change of curvature
strain components:
{e} =
-w.xx
V,y
>+<
/ 2
> + z<
-W,yy
U,y+V,x^
W,yj
= {Em} + {sb} + z{K}
(1)
where and {eb}are the membrane strain components due to in-plane
displacements u and v and the transverse deflection w, respectively. The stress
resultants, membrane force {N} and bending moment {M}, are related to the strain
components as follows:
■[A] IB]'
[B1 [Di;
(2)
where [A] is the elastic extensional matrix, [D] is the flexural rigidity matrix, and
[B] is the extension coupling matrix of the laminated plate.
Element Displacements, Matrices and Equations
Proceeding from this point, the displacements in equation (1) are approximated
over a typical plate element , e.g. rectangular [8] or triangular [9], using the
corresponding interpolation functions. The in-plane displacements and the linear
strains are interpolated from nodal values by
1376
where [Hm] and [Bm] denote the displacement and strain interpolation matrices,
respectively, and {Wm} is the in-plane nodal displacement vector. The transverse
displacement, slopes and curvatures are interpolated from the nodal values by
w = [Hb]{w^}, |^’^| = [G]{wb}, {K} = [B|,]{wb} (4a,b,c)
where [HJ and [G] and [BJ denote the bending displacement, slope and curvature
interpolation matrices, respectively, and {wb } denotes the nodal transverse
displacements and its derivatives. Through the use of Hamilton’s principle, the
equations of motion for a plate element undergoing large amplitude vibration may
be written in the form
[hb]
0
T [H]
[kB]l
r[klNm]+[klNB] [klbml
,r[k2b] <
[k»]J
L ^'"*1 0 J'
[ 0 (
or
[m]{ w} + m + [kl] + [k2] }{w} = 0 (5)
where [m] and [k] are constant matrices representing the element mass and linear
stiffiaess characteristics, respectively; Pel] and Pe2] are the first order and second
order non-linear sfiffiiess matrices, respectively; PcInJ depends linearly on
unknown membrane displacement ({Nm }= [A][Bni]{wni}); PcInb] depends linearly
on the unknown transverse displacement ({NB}=[B][Bb]{Wb}); DelbnJ depends
linearly on the unknown plate slopes and represents coupling between membrane
and bending displacements; and [k2b] depends quadratically on the unknown plate
slopes.
System Equations
After assembling the individual finite elements for the complete plate and
applying the kinematic boundary conditions, the finite element system equations of
motion for the large-amplitude free vibration of a thin laminated composite plate
can be expressed as
[M]{w}4- ([K]+ [K1(W)] + [K2(W)]){W} = 0 (6)
where [M] and [K] are constant matrices and represent the system mass and
stifiBness respectively; and [Kl] and [K2] are the first and second order nonlinear
stifi&iess matrices and depend linearly and quadratically on the unknown structural
1377
nodal displacements { W}, respectively. Most of the finite element large amplitude
fi-ee vibration results for plates and beams in the literature, e.g. references [1,6] and
others, were based on eq. (6) using an iterative scheme and various approximate
procedures. The system equations are not suitable for direct numerical integration
because: a) the nonlinear stif&iess matrices [Kl] and [K2] are functions of the
unknown nodal displacements, and b) the number of degrees of fireedom (DOF) of
the system nodal displacements {W} is usually too large. Therefore, eq. (6) has to
be transformed into modal or generalized coordinates followed by a reduction of
the number of DOF. In addition, the general DufiSng-type modal equations will
have constant nonlinear modal stifihiess matrices. This is accomplished by a modal
transformation and truncation
r=l
where and cou are the natural mode (normalized with the maximum
component to unity) and linear firequency from the eigen-solution ©|_j.[M]{
(|))«=[K]{ (|)}®.
The nonlinear stififtiess matrices [Kl] and [K2] in eq. (6) can now be expressed
as the sum of the products of modal coordinates and nonlinear modal stiffaess
matrices as
[Kl] = ^q,(t)[Kl((])«)] (8)
r=I
n n
[K2] = ^ ^ qr (t)qs (t)[K2((|)('') , )] (9)
r=l s=l
The nonlinear modal stifihiess matrices [Kl]^"^ and [O]^'^^ are assembled from the
element nonlinear modal stif&iess terms [kl]^^^ and Pc2]^“^ as
([K1]«.[K2]‘“))= ^([klf>,[k2]'“>) (10)
al! elements
+ bdy. conds.
where the element nonlinear modal stif&iess matrices are evaluated with the known
linear mode Thus, the nonlinear modal stiffiiess [Kl]^^^ and [K2]^'®^ are constant
matrices. Equation (6) is thus transformed to the general Duf6ng-type modal
equations as
1378
(11)
[M]{q}+([K]+[KlJ+[K2„]){q} = 0
where the modal mass and linear stifhiess matrices are diagonal
([M],[K]) = [<I.f([M],[K])[<I.]
and the quadratic and cubic terms are
LI
[Kl,]{q} = [<Df J^qjKl]
1^)
Vt=1
n n
(rs)
r=:l s=l
\Wk}
(12)
(13)
(14)
AH modal matrices in eq. (11) are constant matrices. With given initial
conditions, the modal coordinate responses {q} can be determined from eq. (11)
using any direct numerical integration scheme such as the Runge-Kutta or
Newmark-P method. Therefore, no updating of the vibration modes is needed [10].
For periodic plate osciHations have the same period T, the response of all modal
coordinates should also have the same period T. Since the initial conditions wHl
affect greatly the modal response, the determination of initial conditions for periodic
plate osciHations is to relate each of the rest modal coordinates in powers of the
dominated coordinate as
arqi(t;IC) + brqi(t;IC) + Crq^(t;IC)+ . = qr(t;IC), r = 2,3. ...n
(15)
where the 2k, br, Cr , . are constants to be determined, and IC denotes initial
conditions. For a three-mode (n=3) system, it is accurate enough to keep up to the
cubic term only in eq. (15) and this leads to two set of equations
a2qi(tp;A,B,C) + b2qf(tp;A,B,C) + C2qJ(tp;A,B,C) = q2(tp;A,B,C), p = 1,2,3
a3qi (tp ; A, B, C) + bsq J (t p ; A, B. C) + Csq J (tp ; A. B, C) = q3 (tp ; A, B, C), p = 1,2,3
(16a,b)
in which the modal coordinates qi, q2 and qs at tp are known quantities and the
initial conditions are qi(0)=A, q2(0) =B, q3(0)=C and qi(0) = q2(0) = q3(0) = 0.
PracticaHy, only eight equations are needed to determine the eight unknowns a2, as.
1379
hi, b3, C2, C3, B and C through an iterative scheme. However, the number of
equations can be more than the number of unknowns for accurate determination of
initial conditions and the least square method is employed in this case.
The time history of the plate maximum deflection can be obtained from eq. (7).
The participation value from the r th linear mode to the total deflection is defined
as
maxjqj.
n
^max|qi
i=l
(17)
Thus, the minimum number of the linear modes for an accurate and converged
frequency solution can be determined based on the modal participation values.
Results and Discussions
Assessment of Single-Mode Elliptic Function Solution
The fundamental frequency ratio Co/col = 2.3051 at Wmax/r =5.0 for a simply
supported beam obtained by Woinowsky-Kiieger [2] using a single-mode and
elliptic fimetion is assessed first. The conventional beam element having six (four
bending and two axial) DOF is used. A half-beam is modeled with 10, 15, 20
elements, and the lowest four symmetrical linear modes are used in the Duffing
modal equations. Table 1 shows that a 20-element and 1-mode model will yield a
converged result. The percentages of participation from each mode for various
values of Wmax/r are given in Table 2. The modal participation values demonstrate
that a single mode (n=l) will yield an accurate fundamental frequency because the
contribution from higher linear modes to the total deflection is negligible (< 0.01 %
for Wmax/r up to 5.0). There is a small difference in frequency ratios between the
present finite element and the elliptic integral solutions. This is due to the difference
between the axial forces of the two approaches, the finite element method (FEM)
gives a non-uniform axial force in each element; however, the average value of the
axial force for each element is the same as the one in the classic continuum
approach. The lowest (2.0310) and the highest (3.3438) frequency ratios at Wmax/r
=5.0 in reference [5] are not accurate.
Frequency ratios for higher modes of the simply supported beam are obtained
next. A model with 40-elements and 3-anti-symmetric modes for the whole beam is
employed for the frequency ratio of the second nonlinear mode. The mode
participations shown in Table 2 indicate that a single-mode approach will yield
accurate frequency results. And the frequency ratios for the second mode are the
same as those of the fundamental one. Thus, the present method agrees extremely
well with Woinowsky-Krieger’s classic single-mode approach.
1380
The time history of the first two symmetric modal coordinates and the beam
central displacement, phase plot, and power spectral density (PSD) at maximum
beam deflection W^Jr = 5.0 for the fimdamental firequency (or mode) are shown in
Fig. 1. The time scale is non-dimensional and Ti is the period of the fundamental
linear resonance. It is noted that although the central displacement response looks
like a simple harmonic motion, it does have a small deviation fi:om pure harmonic
motion due to the second small peak in the spectrum. This is in agreement with
classical solution that the ratio of the jfrequency of the second small peak to that of
the first dominant peak is 3.
Now we are ready to assess the single-mode fundamental firequency of a simply
supported square plate obtained by Chu and Herrmann [3]. A quarter of the plate is
modeled with 6 x 6, 7 x 7, 8 x 8 and 9 x 9 mesh sizes and 1, 2, 4 or 5 symmetrical
modes are used. The conforming rectangular plate element with 24 (16 bending
and 8 membrane ) DOF is used. The in-plane boundary conditions are u = v = 0 on
all four edges. Table 3 shows that the 8 x 8 mesh size in a quarter-plate and 4-
mode model should be used for a converged and accurate frequency solution. Table
4 shows the fi:equency ratios and modal participation values for the lowest three
modes at various Wmax/h for a simply supported square plate (8x8 mesh size in a
quarter-plate). It indicated that at least two linear modes are needed for an accurate
frequency prediction at Wmax/h =1.0, and the contribution of higher linear modes
increase with the increase of plate deflections. The modal participation values also
show that the combined modes (1,3)-(3,1) and (2,4)-(4,2) are independent of the
large-amplitude vibrations dominated by (1,1) and (2,2) modes, respectively. The
time history, phase plot, and PSD at the maximum deflection Wmax/h =1.0 for the
fundamental mode are shown in Fig. 2a, and Tu is the period of the fundamental
linear resonance. There is one small peak in the spectrum and the frequency ratio of
the second small peak to the first dominant one is 3. The low (1.2967) and the high
(1.5314) frequency ratios at Wmax/h =1.0 given in reference [6] are not accurate.
The influence of the initial conditions on periodic motion is demonstrated in Fig.
2a and 2b. In Fig. 2a, the modal coordinates all have the same period, and the initial
conditions are determined fi*om eq.(15). They are qii(0)/h=1.0, qi3+3i(0)/h= -
0.0155, qi3.3i (0)/h = 0.0, q33(0)/h=0.000813, and qi5+5i(0)/h= 0.00011, and initial
velocities are null, whereas in Fig. 2b, qii(0)/h=1.0 and all others are nuU. The
modal coordinates do not have the same period.
Clamped Beam
It is thus curious to find out whether multiple-mode is required for the clamped
beam. Convergence study of the fundamental firequency ratios at Wmax/r =3.0 and
5.0 shown in Table 5 indicates that a 25-element (half-beam) and 4-mode model
win yield accurate and converged results. The time history, phase plot and PSD at
Wmax/r =5.0 are shown in Fig. 3. The modal participation values in Table 6 and the
PSD in Fig. 3 confirm that at least two modes are needed for accurate firequency
results.
1381
Symmetric Composite Plate
A simply supported eight-layer symmetrically laminated (0/45/-45/90)s
composite plate with an aspect ratio of 2 is investigated. The graphite/epoxy
material properties are as follows; Young’s moduli Ei = 155 GPa, E2 = 8.07 GPa,
shear modulus Gn = 4.55GPa, Poisson’s ratio V12 =0.22, and mass density p =
1550 kg/m^ A 12 X 12 mesh is used to model the plate. The in-plane boundary
conditions are fixed (u=v=0) at all four edges. The first seven linear modes are used
as the modal coordinates. Table 7 gives the fundamental firequency ratios and mode
participation values for the linear modes in increasing firequency order. The modal
participation values indicate clearly that four modes are needed in predicting the
nonlinear fi-equency, and other three of the seven are independent of the
fundamental nonlinear mode. Figure 4 shows the time-history, phase plot, and PSD
at Wmax/h =1.0.
UNS YMMETRIC COMPOSITE PLATE
A simply supported two-layer laminated (0/90) composite plate of 15 x 12 x
0.048 in. (38 x 30 x 0.12 cm) is investigated. The graphite/epoxy material
properties are the same as those of the symmetric composite plate. A 12 x 12 mesh
is used to model the plate. The in-plane boundary conditions are fixed at all four
edges. The first four linear modes are used as the modal coordinates. Table 8 gives
the fundamental firequency ratios and mode participation values for the linear modes
in increasing fi-equency order. From the phase plot, the time histories and PSD
shown in Fig. 5, it can be seen that the total displacement response has a non-zero
mean (i.e. the positive and negative displacement amplitudes for all modal
coordinates are not equal). The quasi-ellipse in the phase plot is not symmetrical
about the vertical velocity-axis. In the PSD at Wmax/h =1.0, it is observed that there
are four small peaks in the spectrum and the firequency ratios of the second, third,
fourth and fifth peak to the first dominant one are 2, 3, 4 and 5, respectively. This
observation indicates that the displacement response includes the
superharmonances of orders 2, 3, 4, and 5. The curves, which the positive and
negative displacement amplitudes are plotted against the fundamental firequency
ratio, are also given in Fig. 5. The difference between the positive and negative
amplitudes increases as the firequency ratio increasing.
Conclusions
A multimode time-domain formulation, based on the finite element method, is
presented for nonlinear firee vibration of composite plates. The use of FEM enables
the present formulation to deal with composite plates of complex geometries and
boundary conditions, and the use of the modal coordinate transformation enables to
reduce the number of ordinary nonlinear differential modal equations to a much
smaller one. The present procedure is able to obtain the general Duffing-type modal
equations easily. Initial conditions for all modal coordinates having the same time
1382
period are presented. The participation value of the linear mode to the nonlinear
deflection is quantified ; they can clearly determine the minimum number of linear
modes needed for accurate nonlinear frequency results.
The present fundamental nonlinear fi-equency ratios have been compared with
the single-mode solution obtained by Woinowsky-Kneger for simply supported
beams and by Chu and Herrmann for simply supported square plates. The
Woinowsky-Krieger’s single-mode solution is accurate. For all other solutions,
however, two or more modes are needed. The nonlinear firequencies for
symmetrically and unsymmetrically laminated rectangular composite plates are also
obtained. The phase plot and power spectral density showed that nonlinear
displacement responses are no longer harmonic, and multiple modes are required
for isotropic clamped beams and isotropic and composite plates.
References
1. M. Sathyamoorthy 1987 Applied Mechanics Review 40, 1553-1561. Nonlinear
vibration analysis of plates: A review and survey of current developments.
2. S. Woinowsky-Kreger 1950 Journal of Applied Mechanics 17, 35-36. The effect of an
axial force on the vibration of hinged bars.
3. H. N. Chu and G. Herrmann 1956 Journal of Applied Mechanics 23, 523-540.
Influence of large amplitudes on jfree flexural vibrations of rectangular elastic plates.
4. G. Singh, A. k. Sharma and G. V. Rao 1990 Journal of Sound and Vibration 142, 77-
85. Large amplitude free vibration of beams-discussion of various formulations and
assumptions.
5. H. R. Srirangaraja 1994 Journal of Sound and Vibration 175, 425-427. Nonlinear free
vibrations of uniform beams.
6. S. R. Rao, A. H. Sheikh and M. Mukhopadhyay 1993 Journal of the Acoustical
Society of America 93 (6), 3250-3257. Large-amplitude finite element flexural
vibration of plates/stiffened plates.
7. Y. Shi and C. Mei 1996 Journal Sound and Vibration 193, 453-464. A finite element
time domain modal formulation for large amplitude free vibration of beams and plates.
8. K. Bogna:, R. L. Fox and L. A. Schmit 1966 Proceeding of Conference on Matrix
Methods in Structural Mechanics, AFFDL-TR-66-80, Wright-Patterson Air Force
Base, Ohio, October 1965, 397-444. The gena-ation of interelement compatible
stiffness and mass matrix using the interpolation formulas.
9. Teseller and T. J. R. Hughes 1985 Computer Methods in Applied Mechanics and
Engineering 50, 71 -101 . A three node Mindlin plate element with improved fransverse
shear.
10. A. K. Noor 1981 Composites and Structures, 13, 31-44. Recent advances in reduction
methods for nonlinear problems.
1383
Table 1 . Convergence of the fundamental frequency ratio at Wmax/r =5.0 for a
simply supported beam
No. of elements
and 4 modes
(C0/(0l)i
No. of modes
and 20 elements
(G)/G)l)i
10
2.3537
1
2.3506
15
2.3511
2
2.3506
20
2.3506
3
2.3506
--
4
2.3506
Table 2 The lowest two frequency ratios and the modal participations for a
simply supported beam
W^nax/r
Elliptic
integral [2]
(CO/OOl)!
FEM
Modal
Participation %
(CO/COl)!
_ _
0.2
1.0038
1.0038
100.00
0.000
0.000
0.4
1.0150
1.0149
100.00
0.000
0.000
0.6
1.0331
1.0331
100.00
0.000
0.000
0.8
1.0580
1.0581
100.00
0.000
0.000
1
1.0892
1.0892
100.00
0.000
0.000
2
1.3178
1.3179
100.00
0.002
0.000
3
1.6257
1.6258
100.00
0.004
0.000
4
1.9760
1.9761
99.99
0.005
0.000
5
2.3501
2.3506
99.99
0.009
0.000
W.„ax/r
(co/g)l)2
(C0/CDl)2
q.2_ _
q4
Q6
0.2
1.0038
1.0038
100.00
0.000
0.000
0.4
1.0150
1.0149
100.00
0.000
0.000
0.6
1.0331
1.0332
100.00
0.000
0.001
0.8
1.0580
1.0582
100.00
0.000
0.001
1
1.0892
1.0893
100.00
0.000
0.002
2
1.3178
1.3181
99.99
0.000
0.006
3
1.6257
1.6260
99.98
0.000
0.015
4
1.9760
1.9768
99.98
0.000
0.021
5
2.3501
2.3512
99.96
0.000
0.037
Table 3. Convergence of the fundamental frequency ratios for a simply
supported square plate (Poisson’s ratio=0.3)
Mesh sizes
and
(CD/Qjn
at
W^,Jh
No. of modes
and
(co/coUn
at
W^Jh
4 modes
1.0
1.4
8x8 mesh
1.0
1.4
6x6
1.4174
1.7423
1
1.7028
7x7
1.4163
1.7396
2
1.4169
1,7433
8x8
1.4164
1.7403
4
1.4164
1.7403
9x9
1.4164
1.7400
5
1.4163
1.7401
Table 4. The lowest three frequency ratios and the modal participations for a
simply supported square plate (Poisson’s ratio=0.3)
Elliptic
FEM
integral [3]
(CO/ffiiJii
(to/oiOii
Modal
Participation
%
qu
qi3 -1- qsi
qi3 - qsi
q33
qi5 -f. qsi
0.2
1.0195
1.0195
99.93
0.07
0.00
0.00
0.00
0.4
1.0757
1.0765
99.72
0.27
0.00
0.01
0.00
0.6
1.1625
1.1658
99.38
0.59
0.00
0.02
0.00
0.8
1.2734
1.2796
98.93
1.02
0.00
0.05
0.01
1.0
1.4024
1.4163
98.34
1.57
0.00
0.08
0.01
1.2
1.5448
1.5659
97.54
2.30
0.00
0.15
0.01
1.4
1,6933
1.7401
96.29
3.42
0.00
0.27
0.02
(C0/®l)21
q2i
q23
q4i
q43
...
0.2
N/A
1.0243
99.93
0.06
0.01
0.00
...
0.4
N/A
1.0976
99.50
0.45
0.03
0.01
—
0.6
N/A
1.2072
98.15
1.28
0.54
0.02
...
0.8
N/A
1.3411
97.54
2.41
0.00
0.05
—
1.0
N/A
1.5126
96.24
3.69
0.00
0.08
...
1.2
N/A
1.6900
94.90
4.92
0.03
0.15
...
1.4
N/A
1.8952
93.54
6.01
0.01
0.44
...
y/raJh
(C0/0)l)22
q22
q24 + q42
q24 - q42
q44
...
0.2
N/A
1.0245
100.00
0.00
0.00
...
0.4
N/A
1.0751
100.00
0.00
0.00
0.00
...
0.6
N/A
1.1611
99.99
0.00
0.00
0.01
...
0.8
N/A
1.2806
99.98
0.01
0.00
0.01
...
1.0
N/A
1.4041
99.93
0.01
0.00
0.06
—
1.2
N/A
1.5551
99.97
0.01
0.00
0.01
...
1.4
N/A
1.7074
99.98
0.02
0.00
0.00
—
Table 5, Convergence of the fundamental frequency ratios for a clamped beam
No. of
elements
and 4 modes
(co/©l)i
at
W™/r
No. of modes
and
(©/©lOi
at
Wn.ax/r
3.0
5.0
25 elements
3.0
5.0
10
1.1751
1.4046
1
15
1.1740
1.4009
2
20
1.1732
1.3999
3
25
1.1731
1.3996
4
1385
Table 6 The fundamental frequency ratios and the modal participations for a
clamped beam
Elliptic
integral
(0)/(0l)i
FEM
(ffl/(0L)l
Modal Participation
%
qi
q3
qs
_
1.0222
1.0222
99.78
0.20
0.02
0.00
2.0
1.0857
1.0841
99.33
0.58
0.08
0.02
3.0
1.1831
1.1731
98.35
1.44
0.17
0.04
4.0
1.3064
1.2817
97.37
2.28
0.29
0.07
5.0
1.4488
1.3996
96.26
3.22
0.42
0.11
Table 7 The fundamental frequency ratios and the modal participations for a
simply supported rectangular (0/45/-45/90)s composite plate (a/b=2)
WmJh
Modal
Participation
%
(ffl/coOn
qu
qi2
q2i
qi3
q22
q23
qsi
0.2
1.0408
99.51
0.00
0.00
0.41
0.07
0.00
0.02
0.4
1.1490
96.57
0.00
0.00
3.01
0.24
0.00
0.17
0.6
1.3484
92.93
0.00
0.00
4.55
0.47
0.00
2.04
0.8
1.5241
98.51
0.00
0.00
0.53
0.94
0.00
0.02
1.0
1.7190
97.43
0.00
0.00
2.39
0.11
0.00
0.07
1.2
1.9258
95.78
0.00
0.00
3.57
0.62
0.00
0.02
1.4
2.1409
94.27
0.00
0.00
4.84
0.77
0.00
0.13
Table 8
The fundamental frequency ratios and the modal participations for a
simply supported rectangular (0/90) composite plate
W„,Jh
Modal Participation
%
(CO/COiOn
qii
qi3
qsi
q33
0.2
1.0358
97.83
1.18
0.82
0.18
0.4
1.1432
95.13
2.25
2.24
0.38
0.6
1.2993
94.53
3.86
1.18
0.60
0.8
1.5432
88.56
4.36
4.77
2.31
1.0
1.7880
89.15
3.31
5.06
2.48
1.2
2.0142
92.22
2.89
3.15
1.74
1.4
2.2823
92.01
2.92
2.92
2.15
1386
0.5 1 1.5 2 -6 -3
Time Ratio (tn‘1)
4 6 8 10 12
Frequency Ratio
U
Displacement
Figure 1. Time histories, phase plot and PSD for the fundamental
mode at =5.0 of a simply supported beam
Power Spectrum Density Total OlsplacemonllThIckness qSSn'hJcknBSS q1 1/ThIckness
p P P T* Thousandths o o o
Time Ratio (tn"11)
-0.4 -0.2 0 0.2 0.4
Displacement
Figure 2a. Time histories, phase plot and PSD for the fundamental
mode at W^u/h =1.0 of a simply supported square plate
1388
Thousandths
Displacement Spectrum Density Total Displacement/Thickness q22/Thickness q1 IH'hickrtess
0 12 3 4
Time Ratio {t/T11)
0 12 3 4
0 2 4 6 8 10 12
Frequency Ratio
0 12 3 4
Time Ratio {tfTII)
0 12 3 4
Time Ratio (t/TII)
Displacement
Figure 4. Time histories, plot and PSD for the fundamental
mode at Wmax/h =1.0 of a simply supported
(0/45M5/90)s rectangular plate
1391
Power Specirum DonsUy Total Dlsplacemenin'hlckness qSin’hlckness q11/Thlckness
NONLINEAR FORCED VIBRATION OF BEAMS BY THE
HIERARCHICAL FINITE ELEMENT METHOD
P. Ribeiro and M. Petyt
Institute of Sound and Vibration Research, University of Southampton, Southampton
S017 IBJ, UK
Abstract: The hierarchical finite element (HFEM) and harmonic balance
methods are used to derive the equations of motion of beams, in steady-state forced
vibration with large amplitude displacements. These equations are solved by the
Newton and continuation methods. The stability of the obtained solutions is
investigated by studying the evolution of perturbations of the solutions. Additionally, a
method that allows a quick examination of the stability of the solution is presented and
applied. The convergence properties of the HFEM, the influence of the number of
degrees of freedom and of in-plane displacements are discussed. The HFEM results are
compared with experimental results. Symbolic computation is used in the derivation of
the model.
NOTATION
A - extension coefficient
b - width of the beam
B - coupling coefficient
[C] - damping matrix
D - bending coefficient
[D] - Jacobian of {F}
|D| - determinant of [D]
E - Young's modulus
[E] - elastic matrix
{f} - vector of out-of plane shape functions
|F} - vec. of amplitudes of generalised forces
- vector of generalised forces
{F} - vector of dynamic forces
{g} - vector of in-plane shape functions
h - length of the finite elements
h - thickness of the beam
[Klb] - linear bending stiffiiess matrix
[Kip] - linear stretching stiffiiess matrix
[K2], [K3] and [K4] - components of
nonlinear stiffiiess matrix
[Knl] - nonlinear stiffiiess matrix
L - length of the beam
[M] - mass matrix
[Mb] - bending mass matrix
[Mp] - in-plane mass matrix
[N] - matrix of shape functions
[N'''(x)J - row matrix of out-of-pl. sh. f
p j - number of in-plane shape functions
pQ - number of out-of-plane s. funct.
{qp} - in-plane displacement fimction
{qw} - transverse displacement function
r - radius of gyration
t - time
u - in-plane displacement
Ur - generalised in-plane displacements
w - transverse displacement
Wr - generalised out-of-pl. displ.
{wc}, (wj - coef of cosine and sine terms
X - axial coordinate of the beam
a - loss factor
p - damping factor
eg ,s§ - linear membrane and bending strains
- geometrically nonlin. membrane strain
{si }, {£2 }- linear and geom. nonl. strains
5 W - virtual work of the external forces
cx
SWy - virtual work of the internal forces
5W- - virtual work of the inertia forces
in
X - characteristic exponent
p - mass density
CO - angular fi:equency
cOjj. - natural frequencies
Ja>y - diagonal matrix of squares of
natural firequencies
- Viscous damping ratio
1393
1- INTRODUCTION
In real systems, due to large amplitudes of the excitation, small stiffiiess or
excitation with a frequency in the neighbourhood of resonance frequencies, vibrations
with large amplitudes can occur. In this case, the linear theories may not allow a good
representation of the dynamic characteristics of the system.
A typical case study of vibrations in the nonlinear regime is the forced vibration
of beams, with large displacements. Although a large amount of investigation has been
carried out in this field [1, 2, 10, 18 and others], a method that would allow the
inclusion of higher order mode contributions and damping, without increasing
excessively the number of degrees of freedom (d.o.f ) is desirable. The purpose of this
work is to apply and investigate the advantages of a method that satisfies these
conditions: the hierarchical finite element method (HFEM).
In the HFEM, to achieve better approximations, higher order shape functions
are added to the existing model. Convergence tends to be achieved with far fewer
d.o.f than in the /z-version of the finite element method [4, 11]. The linear matrices
possess the embedding property, meaning that the associated element matrices for a
number of shape functions n=ni are always submatrices for n=n2, n2>ni. The existing
nonlinear matrices of an approximation of lower order, ni, can be used in the derivation
of the nonlinear, matrices of the improved approximation, n2. This makes the construction
of a more accurate model, potentially quicker in the HFEM than in the /z-version.
We are going to consider that the time variation of the solution may be
expressed by harmonics and use the harmonic balance method (HBM). Compared with
perturbation methods, the main advantages of the HBM are its simplicity, the fact that
it is not restricted to weakly non-linear problems and, for smooth systems, the
assurance of convergence to the exact solution [5].
In nonlinear vibrations, the frequency response curves can have multi-valued
regions, turning and bifurcation points. In these regions, we are going to use a
continuation method [8, 14], because, if the Newton method alone is applied, the
solution will depend heavily on the initial guess and convergence is very difficult to
achieve.
Symbolic computation [15] will be utilised, allowing an easier and more
accurate construction of the model.
1394
2 - MATHEMATICAL MODEL
The beam is assumed to be elastic and isotropic, with thin uniform thickness h. The
effects of transverse shear deformations and rotatory inertia are neglected. The transverse
displacement is large compared with the beam thickness, but is very small compared with
the length of the beam (w « L). The slopes are also very small; (w « 1 .
The displacement components u and w may be expressed as the combination of
the hierarchical polynomial shape functions,
§lS2"-Spi ^
0 f,f,..-fn
{d} = |;;) = [N]{q}.[N] =
{q}"=[U, ... U,, W. ... Wj. (2.1)
The hierarchical polynomials used in this study were derived from Rodrigue's form
of Legendre polynomials [4]. Only one element was used to model the whole beam and
only the shape functions that satisfy the boundary conditions were included in the model.
Applying the theory of Bemoulli-Euler, expressing the strain ;
1 as
{s} =
(w^f /2
0
P
_JOo
'Oj ■
(2.2)
I 0 j L8“j [oJ -- --
and equating the virtual work of the inertia forces (D’ Alembert principle) to the
virtual work of the external and elastic restoring forces we obtain:
{5q}^{F}-bJJ{5s,}%{5s,r)[E]({e,} + {s,})dL= {5q}^[M]{q}, (2.3)
° ; A. D = £(1, 2^)Ed2 ; [M] = phbJjN]^[N]dL .
[E] =
The stiffness matrices are defined by:
b[{58,}^[E]{s,}dL={5q}"[Kl]{q}.bJj6eT[E]{8,}dL={6qnK2]{q}.
b({5e3r[E]{8,}dL={6q}"[K3]{q}, b[{5e,}"[E]{84dL={5qnK4]{q} . (2.4)
Considering only transverse forces, if P(t) represents a concentrated force acting
at the point x=Xj and (x, t) represents a distributed force, the generalised forces are
{F} = (£p,(x.t){N”(x)}dL + P,(t){N”(x.)}). (2.5)
In real systems energy is dissipated; consequently, damping should be included
in the present model. For a large variety of materials experimental investigations show
1395
that the energy dissipated per cycle is not dependent on the frequency and is
proportional to the square of the amplitude of vibration [12, 13], The corresponding
type of damping is called hysteretic. We will represent it by a matrix proportional to
the mass matrix and frequency dependent:
[C] = i[M] (2.6)
Considering that damping in the beam results only from the action of the linear axial
and bending strains, we have the following equations of motion;
"0
_K3
K4jj
(2.7)
The in-plane inertia can be neglected for slender beams [3] and the damping
contribution due to the axial stress is generally negligible compared to that due to the
bending stress [12], With these approximations and because [K3] = 2[K2]^, ref [4], we
can simplify the equations of motion to obtain:
+ J[M.]{qw} + [Kl.]{q„} + [Knl]{q„} = {F}, (2.8)
[Knl] = [K4] - 2[K2]^[k1,]'‘[K2] . (2.9)
To integrate exactly terms involving shape functions or its derivatives, present
in the stiffriess and mass matrices, symbolic computation was employed, using the
package Maple [15].
If the external excitation is harmonic and if initial conditions are such that no
transient response exists, then {qw(t)} may be expressed, in a first approximation, as:
{qw(t)}={wc}cos(©t)+{Ws}sin(a)t) (2.10)
We are going to insert this equation into the equations of motion (2.8) and
apply the HBM. This method can be easily implemented in a program written with the
symbolic manipulator For that, one defines the command trign using Maple
library of trigonometric functions, trig, in the following way;
trign: =readlih( 'trig/reduce') [15, 17]. trign, thus defined, replaces all nonlinear
trigonometric functions by linear ones^. With the command coeff one selects the terms
in cos(ot) and sin(a3t). In this way, we obtain equations of motion of the form:
1 For example cos^(Q)t)
is replaced by ^cos(a)t)
+ .icos(3cot)
1396
-CO'
M, 0
0 M,
-3M,
KL
(2.11)
{Fj=|0Knl]{q4cos(cDt)dt=gKM.l + iKM.3){wJ+iK^^^ (2.12)
{F,}=||;[KM]{q4smMdt =iKNL2{wJ + gKNLl + ^K^o]{w^ (2.13)
where KNLl is a iunction of { Wc} only, KNL2 is a function ofboth (wc) and (wj and ICNL3
is a function of {Ws} only These three matrices are, as well as Mb and Klb, symmetric.
3 - STUDY OF THE STABILITY OF THE SOLUTIONS
We will study the problem of local stability of the harmonic solution by adding
a small disturbance to the steady state solution
{q} = {q„} + {8qw} (3.1)
and studying how the variation of the solution evolves. If {6q^} dies out with time
then {q^} is stable, if it grows then {q^} is unstable.
Inserting the disturbed solution (3.1) into equation (2.8), expanding the
nonlinear terms into Taylor series around {q„} and ignoring terms of order higher
than {bq^j , we obtain the variational equation:
K]{5q„} + ^K]{6q„} + [KlJ{5q4 + ^Mj^{5q.}=M (3-2)
The coefficients are periodic functions of time. With symbolic
manipulation, they can easily be expanded in a Fourier series. If {q^} is of the form
(2.10) and since is quadratic in {q„}, we have:
3{qw}
^(M|^=[[p.] + [p,]cos(2cst) + [p,]sin(2o)t)] . (3.3)
2 With this formulation, KNL2 must be calculated using 2[N* J{wJ[N*J(w,} , otherwise .iKNL2
should be considered instead of iKNL2 ■
1397
Simplification (3.14) was possible because the damping matrix is, after
transformation into modal coordinates, equal to a scalar matrix.
Now, following Hayashi [7, page 93], we will express the solution of (3.14) in
the form:
|6f| = e^({bi)cos(®t) + {aj}sin(®t)) (3.15)
which should allow us to determine, in a first approximation, the first order simple
unstable region.
Inserting (3.15) into (3.14) and appl3dng the HBM, we find
(X^[l] + X[M,] + [M,]){y = {°} (3.16)
0 2cd[I]
[-2cd[I] 0 J
[Mo] =
where
[M,] =
(3.17)
^ "1 p'"
[BriDjB]
[Bf[D„lB]-L^+ril]1[l]^[a>o/]
(3.18)
To determine the characteristic exponents, X, we transform this system into [16]
0 [I] Ifxl
-[M„] -[M,]JlrJ
(3.19)
where {X} is a vector formed by {bi}, {aj. The values of X are the eigenvalues of
the double size matrix in the previous equation. Bearing in mind that it is the stability
of the variable {5^} in which we are interested we substitute equation (3.15) into
equation (3.13) to obtain
{61} = {{b, } cos(0t) + {a, }sin((Dt)} . (3.20)
If the real part of X^ -
ii
2 ©
is positive for any then the solution is unstable,
otherwise it is stable.
1398
For undamped systems, it was demonstrated in [8], that important conclusions
about the stability of the solutions can be deducted from the determinant of the
Jacobian of {F}. We are going to extend the demonstration to systems with mass
proportional damping.
Applying the derivation rule for composite functions, we obtain the derivatives
of {Fj} and {fJ with respect to {wc} and {wj as follows;
Matrices [I] and [Mo] are symmetric and matrix [Mi] is skew-symmetric. This
means that the eigenvalues of equation (3.16) are either purely imaginary or purely real
[8]. If X is imaginary the solution is always stable; if X is real the stability limit is
defined by
= (3.25)
Inserting (3.25) in (3.16) we arrive at
■[Bf[D„][B]-co^[l] + [o.,/] [Bf[D.][B] + P l|b,l Jol
[Bf[D„][B]-p [BnD,,][B]-<B^[l] + [co,/]JlaJ loj
The matrix in the previous equation is [B]^[D][B], where [D] is the Jacobian
of {F’} with respect to the vector of coefficients of the cosine and sine terms, given by
[D] = d{F}/d
A non-trivial solution of (3.26) exists if
det([Bf [D][B])=0 o |BnDl=0o|Dl=0. (3.28)
The last equivalence is true, because [B] is a non-singular matrix. Thus, we proved that
in the stability limit, the determinant of the Jacobian of (F), |D| , is zero.
|D| is a polynomial in the coefficients {wj and (wj and in o; therefore, it is a
continuous function in those coefficients. All the experimental and numerical analysis
of nonlinear vibration of beams, indicate that the shape of vibration, defined in our
model by {wc} and {w^}, is a continuous function of the amplitude and the frequency
of vibration. Thus, we conclude that |d| varies in a continuous way through the FRF
(fi-equency response function) curve. Hence, if there is a change in its sign between
two consecutive points of the FRF curve, then |D| =0 for a particular point between
these two. In that particular point, the stability limit might have been crossed.
So, a complete study of the first order solution’s stability may be carried out by
determining only the characteristic exponents of the first solution and when JD] changes
sign or when |D| is approximately zero. As jD| is needed in the continuation method and,
when the Newton method is applied, can be easily calculated from [D], this results in
substantial time savings.
4 -APPLICATIONS
A clamped-clamped beam made in an aluminium alloy with the reference 7075-
T6 was analysed. Its material [9] and geometric properties are:
E = 7. 172*10^° N/m2, p = 2800 kg/m^, h=0.002 m, b=0.02 m, L = 0.405 m 3.
For aluminium, a typical value of the loss factor (which is multiplied by the
stiffness matrix) is a=0.01 (C=0.5%), but the measured value in reference [18] was
approximately equal to 0.038 (^sl.9%)^. In order to have the same damping
coefficient for the first mode of vibration, the value of the damping factor is:
P = (»o,^xa. (4.1)
The beam was modelled using the HFEM, as described in section 2. To solve
the system of equations (2. 11), Newton’s method was used in the nonresonant area. In the
vicinity of resonance frequencies it is difficult to obtain convergence by the Newton method
3 Except in the comparison with experimental results, where L-0.406 (value of Wolfe’s clamped-
clamped beam length).
Wolfe did not think that the measured damping ratio was only due to material damping. He also
attributed the obtained value to damping in the joints and to the coil magnet arrangement used to
excite the beam.
1400
and a continuation method was applied [8, 14]. The derivation of the Jacobian matrix
present in both methods was performed symbolically [15].
Because the excitation force will be applied in the middle of the beam and both
the beam and the boundary conditions have symmetric properties, only symmetric out
of-plane shape functions and antisymmetric in-plane shape functions will be used^.
4.1 - Study of convergence with number of shape functions
With four out-of-plane (po) and four in-plane shape functions, convergence of
the value of the first linear natural frequency is achieved (Table 1). This number of
shape functions will be the starting value for our nonlinear analysis. The number of
degrees of freedom of the present damped model is equal to 2xpo.
Table 1 - Natural linear frequencies of the cc beam (rad/s). Mode 1.
Exact
Po=2, pi=2
Po=3, pi=3
Po=4, p=4i
Po=5, Pi=5
396.6
396.613 239
396. 605 011
396. 605 008
396. 605 008
In Figures 1, 2 and 3 we can see the FRFs in the vicinity of the first, third and
fifth mode, obtained when a force P of 0.03 N was applied. Near the first mode there is
no increase in accuracy by using more than four out-of-plane and four in-plane shape
functions (po=4, pi=4). However, for the third mode, as the amplitude of vibration
grows, the results obtained with po=4 and pi=4 depart from the ones obtained with
more shape functions. The FRF curve constructed with po=5 and pi=5 is quite similar
to the coincident FRFs obtained with po=6, pi=6 and with po=7, pi=7. In the
neighbourhood of the fifth mode, convergence seems to be achieved with po=8 and
pi=8.
4.2 - Influence of in-plane displacements
In Figure 4 the FRFs obtained considering and neglecting the in-plane
displacements are compared. As in references [1] and [10], we found that the in-plane
displacements ‘reduce’ the non-linearity, in the sense that the non-linearity caused by
them is of the soft spring type and counterbalances the hard spring type non-lineanty
5 To check if the nonlinearity introduced any coupling and consequent antisymmetric terms in the
response, a model including symmetric and antisymmetric, in- and out-of-plane shape functions was
considered. It was confirmed that, with these boundary conditions and with the one harmonic
representation of the solution’s time dependency, there is no such coupling.
1401
caused by the transverse displacements. This ‘reduction’ of nonlinearity is due, as the
formulation of the nonlinear stiffness matrix - eq. (2.9) - shows, to the effects of in¬
plane deformation on the stiffness of the structure.
4.3 - Study of stability
In Figures 5 and 6 we can see the stability studies carried out in the
neighbourhood of the first and third resonance frequencies, using po=6, pi=6 shape
functions and with an excitation force of amplitude P = 0.03 N. In all cases, |D|
changed sign when the stability condition of the solutions changed.
4.4 - Comparison with experimental results
In Figures 7 and 8 we can see the comparison between the FRF obtained with
the HFEM, using po=6, pj=6 shape functions, and the experimental results [18] when a
force P = 0.134 N is applied in the centre of the beam. Two values were used for the
damping factor: P=0.01g3q, and P=0.038g)o, .
The HFEM provides a FRF with a slope similar to the experimental one around
the resonance frequency. This indicates that the nonlinear stiffness is well represented
by the model.
The turning point corresponding to the largest amplitude of vibration, where the
jump phenomena occurs, obtained with the HFEM, point B, does not match the
experimental one, point A. With the typical value used for the loss factor in aluminium
alloys, a=0.01 (p=0.01oi)oj), the maximum amplitude of vibration was more than double
the one measured. However, the HFEM solutions represented in Figure 7 after point A are
very close. Thus, in a real system, a small perturbation would easily make the shape of
vibration change into an unstable one and a change, or jump, to a stable shape of vibration
at a lower amplitude could be observed before the largest computed amplitude of vibration
was achieved. With the measured loss factor, a=0.038, the largest amplitude of vibration
obtained with the HFEM, was around a half of the measured maximum amplitude.
1402
5 - CONCLUSIONS
The HFEM dynamic model of a beam vibrating with large amplitudes was
constructed with small time expense. This is due to the small number of degrees of freedom with
which convergence is achieved, to the easy way in which the number of d.o.f are reduced,
benefiting from the symmetry properties of the problem, and to the embedding properties^ of
the HFEM.
For the amplitudes of vibration displacement studied, with relatively few d.o.f
the FRF curves were accurately determined until the 5th order mode, inclusive. If
modes of order higher than 5th are to be studied, then the inclusion of more elements
instead of more shape functions should be considered, as shape functions of excessive
high order turn the construction of the matrices quite time consuming. The comparison
with experimental results showed a very good prediction of the slope of the FRF by the
HFEM. The largest amplitude of vibration and the correspondent turning and jump
point, are greatly influenced by the amount of damping used.
Using the flexibility of choosing the shape functions in the HFEM model it was
shown that the in-plane displacements cause a softening-type nonlinearity.
With the continuation method the multi-valued regions of the FRF curves were
completely and automatically described.
To determine the characteristic exponents that establish the stability of the
solution, we solved an eigenvalue problem. Due to the reduced number of degrees of
freedom of the HFEM model this was quickly solved. More important, it was proven
that in the stability limit the determinant |D| is zero. Thus, we only have to determine
the characteristic exponents of the first solution and when there is an indication that
|Dj =0 for a particular point, to check if the stability of the solution changed. This
results in significant time savings.
With symbolic computation, the matrices involved in the HFEM model and the
Jacobian matrix necessary in the continuation and Newton methods, were easily and
exactly derived, thus reducing the numerical errors. Symbolic computation was also
helpful in the application of the HBM.
^ Here we include the HFEM’s advantages in the derivation of the nonlinear stiffness matrix.
1403
REFERENCES
[ 1 ] - Atluri, S., Nonlinear vibrations of a hinged beam including nonlinear effects. Trans, of
the ASMS J. of Apl. Mech., 1973, 40, 121-126.
[ 2 ] - Bermet, J. A. and Eisley, J. G., A multiple-degree-of freedom approach to nonlinear
beam vibrations. J. of the Am. Inst, of Aeronaut, and Astronaut., 1970, 8, 734-739.
[ 3 ] - Cheung, Y. K. and Lau, S. L., Incremental time-space finite strip method for non-linear
structural vibrations. Earthquake Engng. and Struct. Dynamics, 1982, 10, 239-253.
[ 4 ] - Han, W, The Analysis of isotropic and laminated rectangular plates including
geometrical non-linearity using the p-version finite element method^ Ph.D. Thesis, University
of Southampton, Southampton, 1993.
[ 5 ] - Hamdan, M.N. and Burton, T.D., On the steady state response and stability of non¬
linear oscillators using harmonic balance. J. of Sound and Vibr., 1993, 166, 255-266.
[ 6 ]- Stokey, W. F., Shock and Vibration Handbook, Third edition, ed. C. M. Harris,
McGraw-Hill, New York, 1988, p. 7-14.
[ 7 ] - Hayashi, C, Nonlinear Oscillations in Physical Systems, McGraw-Hill, New York,
1964.
[ 8] - Lewandowski, R., Non-linear, steady-state analysis of multispan beams by the finite
element method. Computers and Struct., 1991, 39, 83-93.
[ 9 ] - ASM Committee on Properties of Aluminium Alloys, Properties and Selection of
Metals, Metals Handbook, Vol. 1, 8th edition, ed. T. Lyman, Ohio, 1961, p. 948.
[ 10 ] - Mei, C and Decha-Umphai, A finite element method for non-linear forced vibrations of
beams. J. of Sound and Vibr., 1985, 102, 369-380.
[ 11 ] - Meirovitch, L., Elements of Vibration Analysis, McGraw-Hill, Singapore, 1986.
[ 12 ] - Mentel, T. J., Vibrational energy dissipation at structural support junctions. In
Colloquium on Struct. Damping, ed. E. J. Ruzicka, 1959, pp. 89-116.
[ 13 ] - Petyt, M, Introduction to Finite Element Vibration Analysis, Cambridge University
Press, Cambridge, 1990.
[ 14 ] - Ribeiro, P. and Petyt, M., Study of nonlinear free vibration of beams by the
hierarchical finite element method. ISVR Techmcal Memorandum No. 773, University of
Southampton, Southampton, November 1995.
[ 15 ] - Redfem, Darren, The Maple Handbook, Springer-Verlag, New York, 1994.
[ 16 ] - Takahashi, K., A method of stability analysis for non-linear vibration of beams. J. of
Sound and Vibr., 1979, 67, 43-54.
[ 17 ] - Wang, S. S. and Huseyn, K., Bifurcations and stability properties of nonlinear systems
with symbolic software. Math. Comput. Modelling, 1993, 18, 21-38.
[ 18 ] - Wolfe, Howard, An experimental investigation of nonlinear behaviour of beams and
plates excited to high levels of dynamic response, Ph.D. Thesis, University of Southampton,
Southampton, 1995.
1404
FIGURES
mnxiwi 1
h
0.5-
€f^
().4-
0.3-
. g
0.2-
«
O.I-
. •
O'
340 360 380 4^0 420
0) (rad/s)
1 - FRF in the vicinity of the first mode of vibration. x=0.5
0 po=4, pi=4; □ po=5, pi=5; 0 po=6, pi=6; + po=7, pi=7.
1405
innxlw
h
1
0.5-
9°’
0.4-
t , .
0.3-
0.2
%
0.1
.
.
- * * ‘ .
0^
340
360
380 400 420 440
CO (rad/s)
Figure 4 - FRF with in-plane displacements, po=6, Pi=6 (o), and without
in-plane displacements, po=6, Pi=0 (+). x=05xL.
Figure 5 - Stability study. First mode. x=0.5xL. □ stable solution;
+ unstable solution; po=6, pi=6.
1406
mnxiwi
h
B
2.2-
2-
1.8-
1.6-
1.4-
1,2-
I-
A
0.8
0.6-
0.4'
0.8 1 .2 , 1.4 1.6 1.8
CO/OD„
0
Figure 7 ~ Comparison with experimental results, o HFEM stable,
□ HFEM unstable, Po=6 and pi=6, |3=0,01cOo ,2; + experimental. x=05xL.
mnxiw
h
1-
A
0.8-
*
0.6
B
0.4
O.2J
.
•
*
0.8 0.9 1 1.1
M/Wq
1.2
1.3
Figure 8 - Comparison with experimental results, o HFEM stable,
□ HFEM unstable, po=6 and pi=6, (5=0.038cOo,2; + experimental. x=05xL.
1407
1408
GEOMETRICALLY NONLINEAR DYNAMIC ANALYSIS
OF 3-D BEAM
Kuo Mo Hsiao and Wen Yi Lin
Department of Mechanical Engineering,
National Chiao Tung University,
Hsinchu, Taiwan, Republic of China
ABSTRACT
A co-rotational finite element formulation for the geometrically
nonlinear dynamic analysis of spatial beam with large rotations but
small strain is presented. The deformation nodal forces and inertia
nodal forces are derived by using the d'Alembert principle and the
virtual work principle. The gyroscopic effect is considered here.
The beam element developed here has two nodes with sbc degrees
of freedom per node. Some angular velocity coupling terms, which
are so called gyroscopic forces, are obtained in inertia nodal force.
An incremental-iterative method based on the Newmark direct
integration method and the Newton-Raphson method is employed
here for the solution of the nonlinear dynamic equilibrium
equations. Numerical examples are presented to demonstrate the
acctuacy and efficiency of the proposed method.
INTRODUCTION
In recent years, the nonlinear dynamic behavior of beam
structures, e.g., framed structures, flexible mechanisms, and robot
aims, has been the subject of considerable research. In [1], Hsiao ^d
Jang presented a co-rotational formulation and numerical
procedure for the dynamic analysis of planar beam structures. This
formulation and numerical procedure were proven to be very
effective by numerical examples studied in [1]. However, it is only
limited for planai* beam structures. A general formulation for three
dimensional beam element is not a simple extension of a two
dimensional formulation, because large rotations in three
dimensional analysis are not vector quantities; that is, they do not
comply with the rules of vector operations. In [2] a motion process
of the three dimensional beam element is proposed for the large
displacement and rotation analysis of spatial frames. In [3] a co-
rotational formulation for three-dimensional beam element is
proposed. However, it is only limited for nonlineai' static analysis.
The objective of this study is to present a practical formulation for
the dynamic analysis of three dimensional Euler beam. The
kinematics of the beam element proposed in [3] is adopted here.
1409
The element deformations are determined by the rotation of
element cross section coordinates, which are rigidly tied to element
CTOSS section, relative to the element coordinate system [2, 3]. The
three rotation parameters proposed in [3] are used to determine the
orientation of the element cross section coordinates. In order to
capture the gyroscopic effect, the relation between the time
derivatives of the rotation parameters and the angular velocity and
the angular acceleration is derived here. The beam element
developed here has two nodes with six degrees of freedom per node.
The element nodal forces are conventional forces and moment.
The deformation nodal forces and inertia nodal forces are derived
by using the d'Alembert principle and the virtual work principle in
the current element coordinates. An incremental-iterative method
based on the Newmark direct integration method and the Newton-
Raphson method is employed here for the solution of the
nonlinear dynamic equilibrium equations. Numerical examples
are presented to demonstrate effectiveness of the proposed method.
FINITE ELEMENT FORMULATION
Basic assumptions
The following assumptions are made in the derivation of the
nonlinear behavior: (1) the beam is prismatic and slender, and the
Euler-Bernoulli hypothesis is valid; (2) the centroid and the shear
center of the ctoss section coincide; (3) the unit extension and twist
rate of the centroid axis of the beam element are uniform; (4) the
cross section of the beam element does not deform in its own plane,
and strains within this cross section can be neglected; (5) the out-of¬
plane warping of the cross section is the product of the twist rate of
the beam element and the Saint Venant warping function for a
prismatic beam of the same cross section; (6) the deformations of the
beam element are small.
Coordinate systems
In this paper, a co-rotational total Lagrangian formulation is
adopted. In order to describe the system, following [3], we define
three sets of coordinate systems (see Fig. 1):
(1) A fixed global set of coordinates, Xj(z = 1,2,3); the nodal
coordinates, displacements, and rotations, and the stiffness matrix
of the system are defined in this coordinates.
(2) Element cross section coordinates, rf (f = 1,2,3); a set of element
cross section coordiaates is associated with each cross section of the
beam element. The origin of this coordinate system is rigidly tied to
1410
the shear center of the cross section. The xf axes are chosen to
coincide with the normal of the corresponding cross section and the
X2 and X3 axes are chosen to be the principal directions of the cross
section.
(3) Element coordinates, x, (z = 1,2,3); a set of element coordinates
associated with each element. The origin of this coordinate system
is located at node 1; the xj axis is chosen to pass through two end
nodes of the element, and the X2 and X2 axes are determined from
the orientation of the element cross section coordinates at two end
nodes using the way given in [2]. The deformations and stiffness
matrices of the elements are defined in terms of this coordinates.
In this paper the element deformations are determined by the
rotation of element cross section coordinates relative to this
coordinate system.
Rotation vector and rotation parameters
For convenience of the later discussion, the term 'rotation vector'
is used to represent a finite rotation. Figure 2 shows that a vector
b which as a result of the application of a rotation vector (pa. is
transported to the new position h'. The relation between b and b'
may be expressed as [4]
b' = cos 0b + (1 - cos 0)(a • b) + sin 0(a x b), (1)
where (p is the angle of counterclockwise rotation, and a is the unit
vector along tiie axis of rotation.
In this paper, the s)mbol { } denotes column matrix. Let e,- and ef
(i - 1, 2, 3) denote the unit vectors associated with the x, and xf axes,
respectively. Here, the traid ef in the deformed state is assumed to
be achieved by the application of the following two rotation vectors
to the traid Cj :
e„ = 0„n, = 0it,
where
n = {0, 82! (el + 03/(0! + 0|)V2}
= {0,^2,713},
t = {cos0„, 62,63}.
cos0„=(i-0!-e!)Vl
^ dw{s) . dv(s)
(2,3)
(4)
1411
in which n is the unit vector perpendicular to the vectors ei and
ef, and t is the tangent unit vector of the deformed centroid axis.
Note that ef coincides with t. is the inverse of cosd„. v(s) and
w(s) are the lateral deflections of the centroid axis of the beam
element in the X2 and directions, respectively, and s is the arc
length of the deformed centroid axis.
The rotation vectors e„ and 0^ are determined by (f = l,2,3).
Thus, di are called rotation parameters in this study.
Using Eqs. (l)-(4), the relation between the vectors and ef
(i = 1,2,3) in the element coordinate system may be obtained as
ef = [t, Ri, R2] = Re,-,
Rl = cos^iri + sin0ir2,
R2 = -sin^iri + cos0ir2,
ri = {-03, cos 0„ + (1 - cos 0„ , (1 - cos d„)n2n^h
r2 ={02^(1-cos0„)w2W3,cos0„ +(l-cos0„)n3}, (5)
where R is the so-called rotation matrix.
Let 0 = {01, 02, 03> be the vector of rotation parameters, 36 be the
variation of 0. The traid ef corresponding to 0 may be rotated by a
rotation vector = {3ipi, 3(l>2, <5^3} to reach their new positions
corresponding to 0 + 50 [3]. When 02 and 03 are much smaller
than unity, the relationship between 50 and 5<^ may be
approximated by
r 1
50 =
-03
^2
03/2 -02/2'
1 0
0 1
(6)
If both sides of Eq. (6) is divided by St, the first time derivative of 0
may be expressed by
r 1
0 =
-03
02
03/2 - 02/2-
1 0
0 1
(j) = T^^ij),
(7)
where the symbol ( ) denotes differentiation with respect to time t .
1412
= 1,2,3) denote the angular velocities about the axes.
From Eq. (7), the second time derivative of 6 may be expressed by
may be expressed as
9 = t-^(j) + T"^^, (8)
where = 1,2,3) denote the angular accelerations about the Xi
axes.
Nodal parameters and forces
The global nodal parameters for the system of equations
corresponding to the element local nodes j (j - 1, 2) are ll^j, the Xj
(i = 1,2,3) components of the translation vectors at nodes j, and
the Xi (f = 1,2,3) components of the rotation vectors at
nodes j. Here, the values of are reset to zero at current
configuration. Thus, <50^, the variations of ^ij, represent
infinitesimal rotations about the Xi axes [3], <i>ij and Oy represent
angular velocities and angular accelerations about the Xj axes,
respectively. Tlie generalized nodal forces corresponding to dOy are
the conventional moments about the Xf axes. The generalized
nodal forces corresponding to dllij, the variation of Uy-, are the
forces in the X, directions.
The element employed here has six degrees of freedom per node.
Two sets of element nodal parameters termed 'explicit nodal
parameters' and 'implicit nodal parameters' are employed. The
explicit nodal parameters of the element are used for the assembly
of the system equations from the element equations. Thus, they
should be consistent with the global nodal parameters, and are
chosen to be Uij, the x, (i = l,2,3) components, of the translation
vectors uj at nodes j (j = 1, 2) and 0y, the Xi (i = 1,2,3) components
of the rotation vectors (j)j at nodes j. Similarly, the generalized
nodal forces corresponding to Wy and d^ij are /y and my, the forces
in the Xi directions and the conventional moments about the Xj
axes, respectively.
The implicit nodal parameters of the element are used to
determine the deformation of the beam element. They are chosen
to be My, the Xj (i = l,2,3) components of the translation vectors u^
at nodes j and 0y, the nodal values of the rotation parameters 0-
1413
(i = 1,2,3) at nodes ; (j = 1, 2). The generalized nodal forces
cori'esponding to duij and dOij are and m|, the forces in the Xi
directions and the generalized moments, respectively. Note that
are not conventional moments, because S6ij are not
infinitesimal rotations about the Xi axes.
In view of Eq. (6), the relations between the variation of the
implicit and explicit nodal parameters may be expressed as
dui'
1
0
0
0
dui
dQi
0
TI^
0
0
5<i>i
' dU2
► —
0
0
I
0
5u2
502
0
0
0
Ti\
^^2
(9)
where daj = {duij,du2j,Su3j}, d6j={ddij,5d2j,862j}, and
% ={8<hj’^<p2j>^<p3jh (j = % 2). I and 0 are the identity and zero
matrices of order 3x3, respectively. (j = 1, 2) are nodal values
of T"l
Let f = {fi,mi,f2,m2K f® ={fi,mf,f2,m^}, where ij ={fij,f2jj3jh
and mj (/’ = h 2), denote the
internal nodal force vectors corresponding to the variation of the
explicit and implicit nodal parameters, dq and, <5q^, respectively.
Using the contragradient law [5] and Eq. (7), the relation between f
and, f ^ may be given by
f = (10)
Kinematics of beam element
The deformations of the beam element are described in the current
element coordinate system. From the kinematic assumptions made
in this paper, the deformations of the beam element may be
determined by the displacements of the centroid axis of the beam
element, orientation of the cross section (element cross section
coordinates), and the out-of-plane warping of the cross section [3].
Let Q (Fig. 1) be an arbitrary point in the beam element, and P be the
point corresponding to Q on the centroid axis. The position vector
of point Q undeformed and deformed configurations may be
expressed as
1414
and
r^xei + ije2 + ze^,
(11)
r = + ^(s)e2 + ^(s)e3 + yel + ze| + Si^s^Qv (1^)
where Xf,{s), v{s) and w{s) are the xy X2 and X't, coordinates of point
P, respectively, s is the arc length of the deformed centroid axis
measured from node 1 to point P. ^c(®) expressed by
Xc{s) = Mil +
where un is the displacement of node 1 in the xi direction, and
cos0„ is defined in Eq. (4).
Here, z;(s) and w{s) in Eq. (12) are assumed to be the Hermitian
polynomials of s , and 0i(s) in Eq. (12) is assumed to be linear
polynomials of s, and maybe given by
z;(s) = {Ni,N2,N3,N4}^{W21/%1/W22/%2}-=
w{s) = {N'i,-N2,N3-,N4}^{m31,021/^32^^22} =
0l(s) = {N5,N6}^{0n^%> = N^u^, (14)
Ni = 7(1 - ?)^(2 f I), Nj = |(1 - 1^)(1 - 1).
4 o
yvj = i(i + 1)2(2 - f), Ni = |(-1 + ?2)(1 + 1).
4 o
Af5 = |(l-?). W6=|(l + |), (15)
where S is the arc length of the centroid axis of the beam element
and may be expressed by
S^ltj QOS (17)
where I is the chord length of the centroid axis of the beam
element, and cos is given in Eq. (4).
The way to determine the current element cross section
1415
coordinates at both ends, element coordinates, and element implicit
nodal parameters corresponding to displacement increments is
given in [2, 3].
If ;c, y, and z in Eq, (11) are regarded as the Lagrangian
coordinates, the Green strains en, Si2, and 613 are given by
£11 =
1
2
\dxj [dxj
-1
£12
2[^xJ
2[^xJ \dzj'
(18)
Substituting Eqs. (4), (5), (12), and (13) into Eq. (18), sn, ei2, and £13
can be calculated.
Element nodal force vector
The element nodal force vector (Eq. (10)) corresponding to the
implicit nodal parameters are obtained from the d'Alembert
principle and the virtual work principle. For convenience, the
implicit nodal parameters are divided into four generalized nodal
displacement vectors u,- (i = a,b,c,d), where
Urt «{«!!, «i2}/
and u^, Uc, and u^ are defined in Eq. (14)..
The generalized force vectors corresponding to (5Uj, the variation
of u,- (f = a,h,c,d) are
4 - + 4 =
where f^ and f*- {i = a,b,c,d) are the deformation nodal force vector
and the inertia nodal force vector, respectively.
The virtual work principle requires that
1416
du^Ja + <50^4 + dn^ic +
= JL {oiideii + 2ai2dei2 + 2ai3(5ei3 + p8r^r)dV ,
where on = Esiy O12 = 2Gei2 and 0-13 = 2Gei2, where E is tl^
Young's modulus and G is shear modulus, p is the density, and V
is the volume of the undeformed beam.
If the element size is properly chosen, the values of the nodal
parameters (displacements and rotations) of the element defined in
the current element coordinate system, which are the total
deformational displacements and rotations, may always be much
smaller than unity. Thus only the first order terms of nodal
parameters are retained in deformation nodal forces. However, in
order to include the effect of axial force on the lateral forces, a
second order term of nodal parameters is retained. Because the
values of the nodal parameters of the element may always be much
smaller than unity, it is reasonable to assume that the coupling
between the nodal parameters and their time derivatives are
negligible. Thus only zeroth order terms of nodal parameters are
retained in inertia nodal forces.
From Eqs. (4), (5), and (12)-(21), the deformation nodal forces and
the inertia nodal forces may be expressed as
Li
(22, 23)
. . rd Gj{di2-Qn)f 1 11
f“ = + k^)Uc, td- ^ i
(24, 25)
fa =
(26)
4 = + m^)Ub - 2p4 ,
(27)
fc = - 2ply fN'cdidsds ,
(28)
f^ = m^Urf - p{ly -
(29)
1417
where A is title cross section area, L is the initial length of the beam
element, k,- and k^- {i = b,c) are bending and geometric stiffness
matrices of conventional beam element [5,6], and J is the torsional
constant p is the density, ly and 7^ are the moment of inertia of
the beam cross section about the and axes respectively, m^ is
the consistent mass matrix of bar element for axial translation, nij-f
and m,> {i^b,c) are the consistent mass matrices of elementary
beam element for lateral translation and rotation, respectively, and
is the consistent mass matrix of bar element for axial rotation.
These mass matrices can be found in [5, 6]. The underlined terms in
Eqs. (27)-(29) are inertia forces induced by the gyroscopic effect, and
are called gyroscopic forces.
Element Matrices
The element stiffness matrices and mass matrices may be obtained
by differentiating the element nodal force vectors with respect to
nodal parameters, and time derivatives of nodal parameters.
However, element matrices are used only to obtain predictors and
correctors for incremental solutions of nonlinear equations in this
study. Thus, approximate element matrices can meet these
requirements. The stiffness matrices and mass matrices of
elementary beam element given in [5, 6] are also used here.
Equations of motion
The nonlinear equations of motion may be expressed by
F^ = F^ + F^-P = 0 (30)
where F^ is the unbalanced force among the inertia nodal force F^,
deformation nodal force F^, and the external nodal force P. F^ and
F^ are assembled from the element nodal force vectors in Eq. (10),
which are calculated using Eqs. (10) and (22)-(29) first in the current
element coordinate system, and then transformed from current
element coordinate system to global coordinate system before
assemblage using standard procedure.
APPLICATIONS
An incremental iterative method based on the Newmark direct
integration method and the Newton-Raphson method is employed
here for the solution of the nonlinear d5mamic equilibrium
equations.
1418
The example considered is a right-angle cantilever beam subjected
to an out-of-plane concentrated load as shown in Fig. 3. Four
elements are used for discretization. A time step size of At = 0.25^ is
used. The cantilever undergoes a finite free vibration with
combined bending and torsion after the removal of the applied
load; the time histories of out-of-plane displacements of the elbow
and of the tip are given in Figs. 4 and 5. It is seen that the present
results are in excellent agreement with those given in [7] and [8].
However, it should be mentioned that the beam elements used in
[7] and [8] are derived using fully nonlinear beam theory and total
Lagrangian formulation. Thus, the beam elements used in [7] and
[8] are much more complicated than that proposed here.
CONCLUSIONS
A co-rotationai finite element formulation for the geometrically
nonlinear dynamic analysis of spatial beam with large rotatio^ but
small strain is presented. The deformation nodal forces and inertia
nodal forces are derived by using the d'Alembert principle and the
virtual work principle. The gyroscopic effect are considered here.
The nodal coordinates, displacements, rotations, velocities,
accelerations, and the equation of motion of the system are defined
in a fixed global set of coordinates. The beam element developed
here has two nodes with six degi'ees of fi^eedom per node. The
element nodal forces are conventional forces and moments. AU of
element deformations and element equations are defined in terms
of element coordinates which are constructed at the current
configuration of the beam element. The element deformations are
determined by the rotation of element cross section coordinates,
which are rigidly tied to element cross section, relative to the
element coordinate system. In conjunction with the co-rotational
formulation, the higher order terms of nodal parameters in
element nodal forces are consistently neglected.
An incremental-iterative method based on the Newmark direct
integration method and the Newton-Raphson method is employed
here for the solution of the nonlinear dynamic equilibrium
equations. Numerical examples are presented to demonstrate the
accuracy and efficiency of the proposed method
It is believed that the co-rotational formulation for 3-D beam
element presented here may represent a valuable engineering tool
for the d5mamic analysis of spatial beam structures.
ACKNOWLEDGMENT
The research was sponsored by the National Science Council,
Republic of China, under contract NSC86-2212-E-009-006.
1419
REFERENCES
1. Hsiao, KM. and Jang, J.Y., . Nonlinear dynamic analysis of elastic
frames. Computers & Structures, 1991, 33, 769-781.
2. Hsiao, KM. and Tsay, C.M., A motion process for large
displacement analysis of spatial frames. International Journal of
Space Structures, 1991, 6, 133-139.
3. Hsiao, K.M., Corotational total Lagrangian formulation for
three-dimensional beam element. AIAA Journal, 1992, 30, 797-
804.
4. Goldstein, H., Classical Mechanics, Addision-Wesley, Reading,
MA,1980.
5. Dawe, D.J., Matrix and Finite Element Displacement Analysis of
Structures, Oxford Univ. Press, New York, 1984.
6. Hsiao, I<.M., A study on the dynamic response of spatial beam
structures. NSC 82-0401-E009-081 Report, National Science
Council, Taiwan, Republic of China,, 1993.
7. Simo, J.C. and Vu-Quoc, L., On the dynamics in space of rods
undergoing large motions - A geometrical exact approach.
Computer Methods in Applied Mechanics and Engineering,
1988, 66, 125-161.
8. lura, M. and Atluri, S.N., Dynamic analysis of finitely stretched
and rotated three-dimensional space-curved beams. Computers
& Structures, 1988, 29, 875-889.
Fig. 1 Coordinate systems.
1420
Fig.2 Rotational vector
Material Properties:
EA==10"
EIy = EI^=GJ = 10^
Ap = 1
ply=pl,= 10
Time History of Load:
Fig.3 Right-angle Cantilever beam
0 5 10 15 20 25 30
Time
Fig. 4 Displacements in the X3 direction at point B.
Fig. 5 Displacements in the X3 direction at point A.
1422
Nonlinear Response of Composite Plates to
Harmonic Excitation Using The Finite Element
Time Domain Modal Method
Raymond Y, Y. Lee, Yucheng Shi and Chuh Mei
Department of Aerospace Engineering
Old Dominion University, Norfolk, VA 23529-0247
Abstract
A multimode time domain formulation based on the finite element method for large
amplitude vibrations of thin composite plates subjected to a combined harmonic excitation
and thermal load is presented. By using the modal reduction method, the system equations of
motion in physical coordinates are transformed into the linear modal coordinates and the
sizes of the system matrices are reduced drastically. The reduced system modal equations can
be handled easily with less computational efforts. The jBrequency-maximum deflection
relations of simple harmonic, superharmonic and subharmonic responses are predicted by
choosing suitable initial conditions. The procedure for the selection of the initial conditions is
also presented. A laminated composite plate is studied in great detail. External loadings
considered are harmonic excitations or combined harmonic and thermal loads. The steady
state responses of the linear modal coordinates are presented in details at several frequencies.
Their phase plots, power spectrums and time domain graphs are given and discussed .
Introduction
The increase use of advanced composites as high performance structural
components necessitates accurate prediction methods which reflect their
multilayered anisotropic behavior. Thin laminated composite plates subjected to
severe harmonic lateral loadings are likely to encounter flexural oscillations
having amplitudes of the order a plate thickness. For the prediction of forced
vibration response, the multilayered anisotropic behavior, the complex
boundary conditions, and the complex loading cases such as the present of the
thermal loads make the problem even more difficult. Methods of analysis
dealing with large deflections are thus becoming increasingly important.
Whitney and Leissa [1] have formulated the basic governing equations for
nonlinear vibrations of heterogeneous anisotropic plates in the sense of von
Karman. Based on those equations, a number of classical continuum
1423
approaches exists for the analysis of nonlinear plate behavior. In general, the
Galerkin’s method is used in the spatial domain, where the plate deflection is
expressed in terms of one or more linear vibrational mode shapes; and various
techniques in the temporal domain such as the direct numerical integration,
harmonic balance, incremental harmonic balance, perturbation, and multiple
scales methods, to cite a few, are employed. Excellent collections of classical
continuum solutions and reviews on geometrically nonlinear analysis of
laminated composite elastic plates are given by Chia [2,3] and Sathyamoothy
[4]. The internal resonance of nonlinear systems has been thoroughly
investigated using the multiple scales by Nayfeh and Mook [5]. Most recently,
Wolfe et al. [6] have reviewed various analytical methods and have obtained
experiment data on beams and plates excited sinusoidally or randomly. Most of
the classical continuum solutions of composite plates have been limited to
single-mode approximation. This is due to the difficulties in obtaining the
general Duffing-type multiple-mode equations using the Galerkin’s approach
especially for arbitrarily (unsymmetrically) laminated composite plates with
complex boundary conditions.
The finite element method has proven to be a powerful and versatile
approach for structural problems of complex geometries, boundary conditions,
and loadings. Reddy [7] has reviewed the application of finite element methods
to linear and nonlinear anisotropic composite plate problems. In this paper, the
nonlinear steady state periodic responses of thin rectangular arbitrarily
laminated composite plates excited sinusoidally with or without the presence of
thermal load are presented using the finite element time domain modal method.
A rectangular composite plate is studied in detail.
Formulation
The finite element system equations of motion for large amplitude vibrations
of a thin laminated composite plate can be expressed as
[M]{w}+[C]{w}+([K]-[Knt]+[K1(W)]+[K2(W)]){W} = {P(t)}+{PT}
(1)
where [M], [C], [K], [Knt] , {P(t)} and {Pt} are constant matrices and vectors
and represent the system mass, damping, linear stiffness, thermal effort and
1424
loads, respectively; and [Kl] and [K2] are the first and second order nonlinear
stiffness matrices and depend linearly and quadratically on the unknown
structural nodal displacements {W}, respectively. The derivation of the element
matrices and load vectors and their explicit expressions are referred to
references [8,9].
The system equations of motion presented in eq. (1) are not suitable for
direct numerical integration because: a) the nonlinear stiffness matrices [Kl]
and [K2] are functions of the unknown nodal displacements, and (b) the
number of degrees of freedom (DOF) of the system nodal displacements {W}
is usually too large. Therefore, eq. (1) has to be transformed into the modal or
generalized coordinates of much smaller DOF. Various reduction methods for
nonhnear problems have been summarized in an excellent review article by
Noor [10]. For nonlinear dynamic problems, the base vectors need updating
using the modal methods presented in [10]. In the present formulation, the
forced general Duffing-type modal equations will have constant nonlinear
modal stiffness matrices, therefore updating of the base vectors is not needed.
This is accomplished by a modal transformation and truncation
{W} = ^q,(t){(|)}">=[<I>]{q} (2)
r=l
where the system mode shapes are the solution from the linear eigen-problem
cOf [M]{(j)}^^^ =[K]{({)}^^\ The nonlinear stiffness matrices [Kl] and [K2] in
eq. (1) can now be expressed as the sum of the products of modal coordinates
and nonlinear modal stiffness matrices as
[Kl] = ^qr[Kl«)W)] (3)
r=l
and
[K2] = ^ ^q^qs [K2((j)<">(t)<^’ )] (4)
r=l s=l
The nonlinear modal stiffness matrices [Kl]^"^ and [KZ]^"®^ are assembled from
the element nonlinear modal stiffness [kl]^''^ and [k2]^^^^ as
1425
(5)
([K1]">,[K2]'”>) = ^([kl]® ,[k2]<“>)
all elements
+ bdy. conds.
where the element nonlinear modal stiffness matrices are evaluated with known
system linear mode W. Thus, the nonlinear modal stiffness matrices
and are constant matrices. Equation (1) is thus transformed to the
forced general Duffing-type modal equations as
[M]{q}+[c]{q}+([K]+[Klq]+[K2qq]){q} = {F(t)} (6)
where the modal mass, damping, and linear stiffness matrices are
([m], \c\ [k]) = [OJ'T ([M], [C], [K] - [Kot ])[<I.] (7)
and the quadratic and cubic terms in modal coordinates and the modal force
vector are
[Klq]{q} = [<E.f 2^qr[Kl]«
lr=l
[.K2qq]{q} = [3.]^ ^^q,q,[K2]
[<E>]{q}
Vr=l s=l
(8)
(9)
{F} = [cI>f({P(t)}+{PT}) (10)
AH modal matrices in eq. (6) are constant matrices. With given initial
conditions, the response of modal coordinates {q} can be determined from eq.
(6) with any direct numerical integration scheme such as the Runge-Kutta or
Newmark-P method. Therefore, no updating of the vibration modes is needed.
The following is the description of the selection of the initial conditions for
periodic motions.
With the input of suitable initial conditions, three types of solutions, periodic
or nearly simple harmonic, superharmonic and subharmonic solutions, can be
obtained. The selection of each type solution is based on the solution of the 1-
1426
DOF Duffing equation obtained by using the modal reduction method
described earlier. For example, the system equations of motion of a symmetric
composite plate can be reduced to 1-DOF model (Note: the quadratic term is
gone because the plate is symmetric) as
Mjq,+Crqr+Krqr+K2,q^ =FrSin(C0t) (11)
where Mr, Cr, Kr, K2r and Fr are scalar constants and represent the modal mass,
damping, stiffness and force; co is the forcing frequency and the subscript “r”
denotes the linear modal number. The solution of eq. (11) can be assumed as
=AiCos(cot)+ A3cos(3cot) for the simple harmonic and superharmonic
solutions, then two sets of Ai and A3 can be obtained by the substituting of the
assumed qr into eq.(n). One set is for the simple harmonic solution and the
other set is for the superharmonic solution. Based on these solutions, the initial
displacement of the r-th modal coordinate in eq. (6) is chosen as A1+A3. The
initial velocities and all other initial displacements are zero. Similarly, it is
assumed that qr =Aicos(cot)+ Ai/3Cos(cot/3) for the subharmonic solution.
Then, all those initial conditions can be found by repeating the procedure just
described.
Results and Discussions
A simply supported eight-layer symmetrically laminated (0/45/-45/90)s
composite plate is studied in great details. The plate isofl5xl2x 0.048 in.
(38.1 X 30.5 X 0.122 cm). The inplane boundary conditions are immovable, i.e.
u=v=0 on all four edges. The graphite-epoxy material properties are : Ei=22.5
Msi (155 GPa), E2 =1.17 (8.07), G12 = 0.66 (4.55), V12 = 0.22 and p = 0.1458
X 10'^ ib-sVin^ (1550 kg/m^). The C‘ conforming rectangular plate element is
used in the finite element model and the plate is modeled with 12 x 12 (144
elements) mesh. The element has a total of 24 DOF (16 bending and 8
membrane). The lowest six natural frequencies (cOr, r=l,6) and their
corresponding mode shapes are : coi = 55.46 Hz for (1,1) mode, Oh = 125.736
Hz for (2,1) mode, CO3 = 151.951 Hz for (1,2) mode, CD4 = 216.475 Hz for (2,2)
mode, CO5 =250.585 Hz for (1,3) mode and cOe = 310.774 Hz for (3,1) mode.
1427
Two load cases considered are uniformly distributed harmonic excitation over
the plate with and without the piresence of temperature. A constant modal
damping factor, = C,/(2Mr 0)r), of 0.02 and a four-mode solution are used in
the examples (Only the (1,1), (2,2), (1,3) and (3,1) modes are considered
because the uniformly distributed excitation cannot induce any response of the
(1,2) and (2,1) modes, see Table 7 of [1 1]).
Harmonic Excitations
A uniformly distributed pressure load of the form p(x,y,t) =po sincot is
considered for the forced vibration problem. The force intensity is maintained
at po = 0.00438 psi (30.2 Pa), however, the forcing frequency co is varying in a
wide range from 0 to 4.5 times of the lowest linear natural frequency ©i. The
results are shown in Figs. 1 to 4, where the designations for the total responses
are indicated in Fig. 1 , while the time-histories, phase plots and power spectra
are given in Figs. 2 to 4. To make clear the behavior of the vibration response
at particular frequency, each modal coordinate is depicted for understanding
the simple harmonic, superharmonic and subharmonic response of the nonlinear
system.
Figures 2a-c correspond to the responses of the three forcing frequencies at
0.6, 2.4 and 3.8 times of ©i labeled as (1) to (3) in Fig. 1. It can be seen that
the total response of the centre of the plate is dominated by the first mode (It
should be noted that the centre of the plate has zero contribution from the (2,2)
mode). The frequency responses of the four modal coordinates are composed
of superharmonic frequency components of order 2, 3, 5 .. etc. , as well as the
input driving force frequency. At the frequency of the point (3) in Figs. 2c and
2d, the corresponding time histories of the second, third and fourth modal
coordinates are unsymmetric as that of the first modal coordinate is symmetric.
Hence, the plate is vibrating with a non-zero equilibrium position due to the
unsymmetric responses of the second, third and fourth modes.
In Figs. 3a and 3b, which correspond to the responses of two points (4) and
(5) at © = 2.4 ©1 and 4.2 ©i in Fig. 1, the total response of the centre of the
plate is almost pure simple harmonic at that particular frequency range.
In Fig. 4a, which corresponds to the response of the point (6) at © = 3.8 ©i
in Fig 1, the centre of the plate is mainly composed of subharmonic response of
order 1/3. In the time histories of the four modal coordinates of Fig. 4b, it can
be seen that the subharmonic component in the total response is contributed by
1428
the (1,1) modal coordinate, and the responses of the higher modal coordinates
are pesudo harmonic.
Combined Harmonic and Thermal Loads
In addition to the uniform pressure po sincot, a steady state temperature
change of 2.9Tcr is also applied to the composite plate ( where the buckling
temperature =13.79 °F). The forcing frequency is taken as co = 1.45c0i and
it is kept at that constant frequency of excitation, however, three pressure
intensities at po = 6, 10 and 14 x 10*^ psi are considered. The responses are
shown in Figs. 5a-c. The results are shown after the transient response being
damped out, this is demonstrated by the quasi-steady state time histories in
Figs. 5a and 5c.
When the pressure load is small at 0.006 psi (41.3 Pa), the plate exhibits
small oscillations about one of the thermally buckled positions (Wmax/h
=1.0237) shown in Fig. 5a. With increase of the pressure loading, the amplitude
of vibration increases. Fig. 5b shows the so-called snap-through or oil-canning
phenomenon at po = 0.010 psi (68.9 Pa), the plate behavior is chaotic and has
two potential weUs. With the further increase of the pressure loading, the plate
exhibits large amplitude oscillations through the two buckled positions as
shown in Fig. 5c at po = 0.014 psi (96.4 Pa). The plate motion is periodic at
low and high pressure loads, however, the plate response composes of
superharmonic frequency components of order 2 and 3 at the low pressure and
of order 3 and 5 at the high pressure as shown in the PSD plots.
The substance of the transition of the three distinct plate behaviors, from the
small oscillations into the chaotic motion then into the large amplitude
vibrations with the increase of forcing intensity, is shown in Fig. 6. In the low
pressure range, the plate could also vibrate about the another equally possible
bifurcation buckled position shown with dotted lines.
Conclusion
Based on the finite element method, a multimode time-domain formulation
for nonlinear forced vibration of composite plates is presented. The main
advantage of this method is that the system matrix equation derived from the
FEM can be transformed into a set of general type Duffing equations with
constant system matrices and much smaller DOF. The selections of initial
1429
conditions for the subharmonic, simple harmonic and superharmonic responses
are presented. Through detailed descriptions, the frequency response
characteristics, phase plots, time histories and the power spectrums have been
illustrated for the three types of responses. The responses of a thermally
buckled composite plate under harmonic excitation with fixed forcing
frequency and various amplitudes are also obtained. Snap-through motion is
observed at moderate pressure loads.
References
1. J. M. Whitney and A. W. Leissa 1969 Journal of Applied Mechanics 36, 261-266.
Analysis of heterogeneous anisotropic plates.
2. C. Y. Chia 1988 Applied Mechanics Review 41, 439-451. Geometrically nonlinear
behavior of composite plate: A review.
3. C. Y. Chia 1980 Nonlinear Analysis of Plates, McGraw-Hill, New York.
4. M. Sathyamoorthy 1987 Applied Mechanics Review 40, 1553-1561. Nonlinear
vibration analysis of plates: A review and survey of current developments.
5. A. H. Nayfeh and D. T. Mook 1979 Nonlinear Oscillations, John Wiley, New
York.
6. H. F. Wolfe, C. A. Shroyer, D. L. Brown and L. W. Simmons 1995 Technical
Report WL-TR-96-3057, Wright Laboratory, Wright Patterson AFB, Ohio. An
experimental investigation of nonlinear behavior of beams and plates excited to
high levels of dynamic response.
7. J. N. Reddy 1985 Shock and Vibration Digest 17, 3-8. A review of the literature
on finite element modeling of laminated composite plates.
8. Y. Shi and C. Mei 1996 Proceedings of the 37th AIAA / ASME / ASCE / AHS /
ASC Structures, Structural Dynamics, and Material Conference, Salt Lake City,
UT, 1355-1362. Coexisting thermal postbuckling of composite plates with initial
imperfections using finite element modal method.
9. C. K. Chiang, C. Mei and C. E. Gray, Jr. 1991 Journal of Vibration and Acoustics
113, 309-315. Finite element large-amplitude free and forced vibrations of
rectangular thin composite plates.
10. A K. Noor 1981 Composites and Structures, 13, 31-44. Recent advances in
reduction methods for nonlinear problems.
11. Y. Y. Lee, Y. Shi and C. Mei 1997 Proceedings of the 6th International
Conference on Recent Advances in Structural Dynamics, University of
Southampton, UK. A finite element time domain multi-mode method for large
amplitude free vibration of composite plates.
1430
Centre Disp/Thlckness Centre Disp/Thickness
Po=0.00438 Psi
Damp. Ratio = 0.02
(3)
(2)
(1)
■ ■
(4)
(6)
(5)
0 1 2 oa/osl 3 4 5
Freq. Ratio
Figure 1. Frequency response of the simply supported (0/45/-45/90)s rectangular plate
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
Po (psi)
Figure 6. Plate centre response vs force amplitude at co = 1.45ci>i and T = 2.9 Ter
1431
1432
1433
1434
1435
1436
GEOMETRICALLY NONLINEAR RESPONSE ANALYSIS
OF LAMINATED COMPOSITE PLATES AND SHELLS
C.W.S. Tot and B. Wangt
Department of Mechanical Engineering
University of Nebraska
255 Walter Scott Engineering Center
Lincoln, Nebraska 68588-0656
U.S.A.
E-mail; cwsto@unlinfo.unLedu
Abstract
The investigation reported in this presentation is concerned with the prediction
of geometrically large nonlinear responses of laminated composite plate and
shell structures under dynamic loads by employing the hybrid strain based flat
triangular laminated composite shell finite elements. Large deformation of
finite strain and finite rotation are emphasized. The finite element has eighteen
degrees of freedom which encompass the important drilling degree of freedom
at every node. It is hinged on the first order shear deformable lamination
theory. Various typical laminated composite plate and shell structures under
dynamic loads have been studied and representative ones are presented and
discussed in this paper. Shear locking has not appeared and there is no zero
energy mode detected in the problems studied. It is very accurate and efficient.
Consequently, it is relatively much more attractive than other elements
currently available in the literature for large scale nonlinear dynamic response
analysis of laminated composite plate and shell structures.
t Professor and corresponding author
i Research Associate
1437
1. INTRODUCTION
Many modem structures such as nuclear reactor containment installations,
naval and aerospace structures, and their components, must be designed to
withstand a variety of intensive dynamic disturbances. Because of their many
attractive features over isotropic materials more and more stmctures or
components in the aforementioned systems are made of laminated composite
materials. The investigation reported in this paper is therefore concerned with
the prediction of geometrically large nonlinear responses of laminated
composite plate and shell structures, of complicated geometries, under transient
excitations. With complicated geometries analytical solution is impossible and
therefore a versatile numerical method, the finite element method has been
employed. A hybrid strain based flat triangular laminated composite shell
finite element has been developed by the authors [1,2] for the nonlinear
analysis of plate and shell structures under static loadings. The present
investigation is an extension of [1,2] to cases with the aforementioned
dynamic forces. Among various attractive features of the derived element
stiffness and consistent element mass matrices five are worthy of listing here
for completeness. These are: (a) their ability to deal with large nonlinear
elastic response of finite strain and finite rotation, (b) the fact that they are in
explicit expressions and therefore no numerical integration is necessary, (c) the
obtained results of a relatively comprehensive tests [2, 3] show that the
element is free from shear locking, (d) the element gives correctly six rigid
body modes, and (e) the finite element has three nodes and eighteen degrees
of freedom (dof) which encompass the important drilling degree of freedom
(ddof) at every node. It is based on the first order shear deformable lamination
theory. It is a generalization of the low-order flat triangular shell element for
isotropic materials developed earlier by Liu and To [4].
It is noted that one of the earlier work that employed triangular shell
element is due to Noor and Mathers [5]. In the latter a mixed type triangular
element was proposed. The element has six nodes, and 78 dof. It was based
on the shallow shell theory and was shear deformable. Recently, Lin et al [6]
developed a finite element procedure to analyze composite bridges. The finite
element procedure was based on small elasto-plastic strains and updated
Lagrangian formulation. The element used was flat and constructed by the
superposition of a discrete Kirchhoff bending element and a linear strain
triangular membrane element. It has six nodes. There are three translational
and three rotational dof at its comer nodes and three translational dof at mid¬
side nodes. In 1994 a flat triangular shell element was presented for static
nonlinear analysis by Madenci and Barut [7]. It is based on the so-called free
formulation concept for analyzing geometrically nonlinear thin composite
shells. A corotation form of the updated Lagrangian formulation is utilized.
The theoretical basis was on the geometrically nonlinear Kirchhoff plate theory
without considering the effects of transverse shear deformation. The element
1438
is of displacement type. It has three nodes and six dof for each node. While
such formulation has some advantageous features computationally the element
is relatively less efficient because (a) the linear element stiffness matrix
consists of a basic and a higher-order stiffness matrices in the sense of
Bergan and Nygard [8] for isotropic materials, and (b) the important effects
of transverse shear deformation in the plate component of of this element has
been disregarded. A more recent contribution on triangular elements is made
by Zhu [9]. The natural approach is used to construct a curved triangular shell
element for static analysis of geometrically nonlinear sandwich and composite
shell structures. The element has six nodes. There are six dof at each corner
node and three dof at each mid-side node. Updated Lagrangian description was
adopted in the procedure. In the element formulation the transverse shear
deformation was considered by assuming constant transverse shear stress
distribution.
In the next section the formulation of element stiffness matrices is
outlined. Section 3 deals with the derivation of element matrices. Section 4 is
concerned with the application of the derived elements to three example
problems of plate and shell structures. The concluding remarks are included
in Section 5.
2. FORMULATION OF ELEMENT STIFFNESS MATRICES
Finite element formulation for the derivation of a family of simple
three-node, six dof per node, hybrid strain based laminated composite
triangular shell finite elements for large scale geometrically nonlinear analysis
is briefly outlined in this section. Large deflection of finite strains and finite
rotations are included. The first order shear deformation theory and the
degenerated three dimensional solid concept are adopted. In particular, element
matrices for one member of the family are derived explicitly with the
symbolic computer algebra package MACSYMA. To minimize the algebraic
manipulation involved in the derivation, updated Lagrangian description is
employed in the incremental formulation of the finite element procedure. In
essence, the present formulation is an extension of the work by Liu and To [4]
for isotropic materials to multi-layer laminated composite shells. Therefore, in
the development the present approach follows closely that of the last reference.
2.1 Incremental variational principle
The Hellinger-Reissner functional tChr can be written as
where,
e' is the independently assumed strain field;
e“ is the strain due to displacement;
C is the material stiffness matrix or elasticity matrix;
W is the work done by external forces,
and the superscripts e and u indicate that the quantities are from independently
assumed strain field and displacement field, respectively. For geometrically
nonlinear analysis with incremental formulation and updated Lagrangian
description, the static and kinematic variables in current equilibrium
configuration at time t are assumed to be known quantities and the objective
is to determine their values in the unknown subsequent equilibrium
configuration at time t+At. For a time increment At, that is from time t to
(t+At), one has
- ATTjjj^CAUjAe®) - 7ifjjj(t+At) - ttHR® ,
or, with reference to equation (1),
Atihr " / [(e')TC(Ae“)
- i(Ae‘)fC(Ae') -
where,
Au is the vector of incremental displacement;
Ae* is the vector of independently assumed incremental updated
Green strains;
Ae“ is the vector of incremental updated Green ’geometric’ strains
or incremental Washizu strains;
AW is the work-equivalent term corresponding to prescribed body
forces and surface tractions in configuration C
Equation (3) is the incremental form of Hellinger-Reissner variational
principle. For updated Lagrangian description, the integral is evaluated at the
current configuration C In the equation, the term
J(Ae')^C(e^ - e“) dV
Ve
■f (Ae'')'^C(Ae“)
(3)
(Ae')'^C(e‘^ - e'^)] dV - AW .
1440
is the so-called compatibility-mismatch. Numerical results of Saleeb et al [10]
showed that though totally discarding the term resulted in convergence
difficulties, while including the term in only the first iteration of every load
step yielded essentially the same results as those having the term under all
circumstances. However, Liu and To [4] reported no difficulties for
convergence when the term was ignored. In the current study, this term is also
disregarded. Then equation (3) can be recasted as
^^hr“/ [ + (Ae®)^C(Ae“)
Vc
(4)
-■i(Ae')’'C(Ae')]dV-AW,
2
where = (e^)^C is the Cauchy (true) stress vector at the current
configuration C ^ In this equation, the incremental Washizu strain Ae“ can be
expressed in two parts
Ae^
Aei" ^ ATii“
(5a)
and they are related to the incremental displacement by
|(Auy AUjj) . Arii”
-AUyAU,^
(5b, c)
where the Einstein summation convention for indices has been adopted and the
differentiation is with respect to reference co-ordinates at the current
configuration C
Substituting equation (5a) into (4) yields
A’'™ = f [- -(Ae')’'C(Ae') + (Ae')'^C(A£") + a'^Ae”
V. 2 (6)
+ o^Aii" + (Ae')fCATi“ ] dV - AW ,
where a is the Cauchy stress vector.
Discarding the higher order term, ) ^Aq ^ results in
AjIhr “ / [- -(Ae')'^C(Ae') + (Ae')TC(AE“)
+ a^Ae'^ + a^Aq^] dV - AW .
(7)
1441
2.2 Hybrid Strain Formulation
Element stiffness matrices for a hybrid strain based finite element can
be derived directly from equation (7). Generally the independently assumed
strain field and displacement field can be written as
Ae' = P Att , Au = <t) Aq
where P is the strain distribution matrix, (|) is the displacement shape function
matrix, Aa is the vector of incremental strain parameters and Aq is the
incremental nodal displacement. Substituting equations (8a, b) into (7), and
defining
H - J P^CP dV^ , G, = / P'^CBl dV^ ,
V - / dVe . F. " / BlO dV. ,
V V,
one can show that
AT:jnj(Aq,Aa) = ’ ^Aa'^HAa + Aa'^G^Aq
+ F^Aq + ■iAq^'k^LAq - F'^Aq ]
(10)
where F is the external nodal force vector in the neighbour configuration
associated with the AW term in equation (7); and B^l are the linear and
nonlinear strain-displacement matrices, while Cq is the matrix containing the
Cauchy stress components at the current configuration.
Finally, one can show that
(k[ . V) Aq - F{t . At) - F, , k[ = g7h-‘G. 0
where the expression in equation (1 lb) is the element "linear" stiffness matrix.
The term k^L defined in equation (9) is the "nonlinear" or initial stress
stiffness matrix and Fi is the pseudo-force vector. The right hand side of
equation (11a) is the equilibrium imbalance.
1442
3. ELEMENT MATRICES AND THEIR UPDATING
The derivation of nonlinear element stiffness matrices, constitutive
equations, mass matrices of the element shown in Figure 1 are outlined here.
In addition, updating of configuration and stresses at every time step is
considered here for completeness.
3.1 Nonlinear Element Stiffness Matrices
For the assumed displacement field, an arbitrary point within an
element is governed by
r*'
3
t
h
t
Si
0
3
(12)
The incremental displacements of an arbitrary point within the element are
Au' 1
1 O
Au/|
3
<
II
AVi 1
> + (AVi') .
i-l
Aw' ,
1
Aw/
1*1
Employing quadratic polynomials for the translational dof and including ddof
lead to
Au' ^
3
AUi'
3
Ai(ii) Ai(i2)
A0'
Av'
Aw'
= E?i^
i-l
AVi'
Awj'
^ c'E^i
i-l
Ai(21) Ai(22)
Ai(31) Ai(32) ^
1-if
0
0
Pi
AO^i
0
0
5i
A
Aei •
. "Pi
-5i
0
Ae'
(14)
1443
In the foregoing,
U = Ui^l + U2^2‘^U3^3 + Piejl + p20t2-"P3®G
W = WiCl+W2^2-"W3^3-Pie,l-P2e ^-536,3
"5i®s1“^2®s2“^3®s3 »
0r = 0rl^l'*'®i2^2-*'®r3^3
®s*®sl^l ■*'®s2^2'^®s3^3
®tl^l '*’^12^2 ®e^3
(15a)
(15b)
where
Pi =■ (^1^3 "^12^2)^! ’ 5i “ (^31^3 “^12^2)^! »
P2 * (^12^1 ”^3^3) ^2 » ^2 * (^12^1 “^23 ^3) ^2 ’
P3 “ (^3^2 ~ ^31 ^1)^3 » ^3 ” (^23^2 1)^3 •
The remaining symbols have been defined by Liu and To [4] and are not
repeated here for brevity.
For the assumed strain field, the strain vector in equation (8a) may be
written as
Ae*^
Ae;
4
Ay'
(16)
where
A^'m = PmA«„ . Ax' - PfcAa,
Ay' - PsA«s
(17a)
1444
with
Aa^-lAttj Aa2 Actj)’'’ ,
(17b)
Aa^-{Aa4Aa5Aa^}'^,Aa^-{Aa7Aa8Aa,}’',
and
1
0
o'
1
0
o'
B
II
0
1
0
11
pf
0
1
0
o,
0
1.
0
0
1
-83(1-2^2) 83(1-2^1) 0
-13(1-2^2) (r3-r2)(l-2^i) 12(1-2^3)
where the subscripts m, b and s denote the membrane, bending and transverse
shear components of P in equation (8).
By defining
= / PjA'P^ da , = f P3"C;P„ da ,
a a
- / Pjc^P. da , - / Ps^E'P, da ,
a a
= / PjB'P^ da , = / P,"CJP, da , (18)
a a
H„b - / PmB'Pb da , P,^CX da .
a a
Hbb = / Pb'D'Pb da
1445
where a is the area of the triangular shell element and
A' - E (C.)k(h,-h,.,) . B' - i ^ (C,)k(h^h,l,) ,
k-1 ^ k-1
" 'T S (^a)k(^k“'^k-l) ’ (^b)k(^k“^k-l) ’
d k-1 k-1
Ca - i: (c j,aik“Vi) > Cb - -i i: (c
k-1 k-1
(19)
in which the integer n is the number of laminae in the laminated composite
structure. Then the matrix H in equation (9) becomes
H =
Hta
Hu
H*
9x9
(20)
Similarly by defining
/ pJa'B^ da .
Gsm - / Ps"cX <ia
a
a
Gms ■
/ da ,
= / Ps^E'B, da ,
a
a
Gbm =
/ P,^B'B„ da ,
Gi, ■= / pXb, da .
a
a
Gmb “
/ pJb'B, da ,
G., - / PsXX ^ .
a a
G,, = / pJd'B, da
1446
one has the matrix
+
G„b
+
G^
B
X)
o
+
Gbb
+
Gbs
Gs.
+
G.b
+
9x18
(22)
Therefore, with the ddof considered the element stiffness matrix can be shown
to be
k - k[
+ k^ + k^L
(23)
where the linear element stiffness matrix k^’ and the "nonlinear" or initial
stress stiffness matrix k^L ^re defined by in equations (9) and (11), while the
stiffness matrix associated with the ddof k^d is defined as
kda = i:(G.),(h,A-.)/B7B,da,
(24)
k-1
in which
Bd = [Bdi Bj2 B^Ji^ig
and
Bdi
-Ui.r 0 0 0
with i = 1,2,3.
The "nonlinear" or initial stress stiffness matrix k^L can be obtained if
the nonlinear strain-displacement matrix B^l and the matrix Gc which contains
the Cauchy stress components at the current configuration are available. The
matrix B^l is defined by equation (45) of Liu and To [4].
The matrix Oq is constructed from the Cauchy stress vector c and
defined as
^11^
°3ll3
^22^3
”23'^
*^23^
O3
(25)
1447
with I3 being the 3x3 identity matrix and O3 a 3x3 null matrix. The transverse
stress components of a are considered constant over the thickness, and all
components of a are calculated and updated for each time step at the centroid
of each element.
3.2 Constitutive Equations
For finite strain problems in the elastic range, the reduced stiffness
matrix is a function of stresses. To incorporate finite strains in the analysis,
several approaches can be applied. The following adopted from reference [4]
is to add the linear elastic matrix a correction matrix which is a function of
Cauchy stress. To begin with, the correction terms in tensor form becomes
Ciid " - * OjiSfl + OnSik + "jiSik )
where 8^^ is the Kronecker delta. Note that this equation comes as a result of
transforming the Jaumann stress rate to the incremental second Piola-Kirchhoff
stress. If the stress and strain vectors are
O = { Ojj O22 O33 0^2 ^23 Ojj } ,
e = { ®22 ®33 ®12 ®23 ®31
the matrix form of equation (26) is
0
0
2012
0
2013
0
4<J22
0
20i2
20,3
0
1
0
0
4033
0
2023
2013
2
20i2
2a,2
0
^ll'^^22
Oi3
^23
0
2°23
2023
°13
^22"^ ^33
^12
2oj3
0
20j3
^^23
^12
Oii + O
In present investigation the so-called degenerated concept is adopted and
therefore the elastic modulus in the normal direction to the plane of the shell
structures is considered zero. Consequently the stress and strain in the
transversal direction are ignored. In the linear analysis the constitutive
relations for a lamina have been defined as
o = Qe (29)
1448
where
°n ° zx ’
e - { e, By e„ )■"
Qn
Qi2
Qi6
0
0
Qi2
Q22
Q26
0
0
Qi6
Q26
0
0
0
0
0
Q44
0
0
0
^5
Q55
The corresponding matrix from equation (28) is
0
2<’.y
0
2°.
0
40y
2<^.y
2o
yz
0
2%
°yy
0
2^.
® zx
°xy
0
^xy
The material stiffness matrix for a lamina thus becomes
C = Q + C
(30)
(31)
(32)
(33)
where
t^aJ3x2
t^aj2x3 J2x2 .
in which C„ C^, and = ^ba are given in equation (19).
With the consideration of large deformation and finite strain, the
constitutive equations for a multilayered structure or laminate can be written
as
1449
or simply
(34)
' N '
■ A'
B'
Ca
^ M
D'
Cb
-
X
Qs
Ub
e'
8x8
. y .
^ ®N
where N, M and Qj are the vectors of stress resultants corresponding to
membrane, bending and transverse shear, respectively. The matrices A’, B’,
D’, E’, and Cg have been defined in equation (19).
3.3 Element Mass Matrices and Updating of Configurations and
Stresses
In the present study, with the updated Lagrangian description, the
consistent mass matrix is formulated in the current configuration C '. The mass
matrix is then updated at each time step. The assumptions are that the angular
velocities and accelerations are small enough to be discarded. By following the
procedures of Liu and To [4] the consistent element mass matrix can be
obtained as
in which mt„ and m„t are translational and rotational components of the
consistent element mass matrix, respectively. Matrix m^ is the part associated
with the ddof. When it is used for the incremental formulation with updated
Lagrangian description, updating relevant quantities at each incremental step
are required before evaluating the mass matrix. All these mass matrices are
obtained explicitly with the symbolic computer algebra package MACSYMA.
For each incremental step, the configuration and stresses have to be
updated. Details of the steps can be found in the reference by Liu and To [4]
and therefore are not included here. However, it may be appropriate to point
out that the linear consistent element matrix for multi-layer composites has
been employed by the authors [11] for vibration analysis of plates and shells.
4. EXAMPLES OF LAMINATED COMPOSITE PLATE AND SHELL
There are two main objectives in this section. First, accuracy of results
obtained by the presently derived element matrices is studied. Second, the
validity and conceptual adequacy of the formulation and assumptions made in
the derivation of element matrices are assessed. For brevity, one multi-layer
plate, one multi-layer shell structure, and a cantilever panel with free end step
moment are included here. More example problems can be found in To and
Wang [3], and Wang and To [12].
4.1 Multi-Layer Plate Under Uniformly Distributed Step Disturbance
The square plate considered has two layers. Its geometrical dimensions
are: side length a = 2.438 m and total thickness h = 0.00635 m. Each layer of
the laminate has equal thickness. The plate stacking scheme^ is cross-ply
(0/90). The layer material properties are: ^ = 6.8974 x N/m^, Ej = 25
Gi2 = G,3 = 0.5 E2, G23 = 0.2 E2, V12 = 0.25 and density p = 2498.61 kg/m .
It is supported by hinges at its four edges. At these edges U or V (note
henceforth upper case of deformation variable refers to global co-ordinate)
parallel to the edges are not constrained. These boundary conditions are
denoted as BCl in reference [13]. For the purpose of direct comparison with
the results reported in the latter reference, one quarter of the plate is modeled
by a 4 X 4 D mesh (see Figure 1 for the definition of D mesh). Thus, the
boundary conditions applied are: V = 0^ = 0.0 at AB, U = W = 0,, = 0.0 at
BC, V = W = 0y = 0.0 at CD and U = 0y = 0.0 at AD. In addition, all 0, are
constrained. After application of the boundary conditions there are 158
unknowns in this case.
The uniformly distributed transversal step disturbance with intensity po
= 490.5 N/m^ is applied to the plate. In the analysis, the option of inclusion
of directors [3, 12] and small strain are selected. The time step size is At =
0.001 seconds. The responses at the centroid obtained by using the HLCTS
element are plotted in Figure 2. They are compared with those reported by
Reddy [13] in which results were obtained with a nine-node rectangular
isoparametric element. In the latter transverse shear was considered. Excellent
agreement can be observed. Before leaving this subsection it may be
appropriate to mention that the nonlinear element stiffness matrix presented
in reference [13] is nonsymmetric while the one derived in the present
investigation is symmetric. In fact, when the system is conservative the
nonlinear element stiffness matrix can be shown to be symmetric.
4.2 Spherical Shell Segment Under A Uniformly Distributed Step
Disturbance
The geometry of the spherical shell is shown in Figure 3 in which the
shell is simply supported. The geometrical properties are: radius R = 10.0 m,
the side length of the projected plane b = 0.9996 m and the total thickness h
= 0.01 m. The spherical shell is considered having two equal thickness layers
and they have the (- 45/45) lamination scheme. The pertinent material
properties are: Ej — 2.5x10^' N/m^ E2 = 1.0x10^° N/m^, Gi2 = G13 = 0.5x10 ^
N/m^, G23 = 0.2x10*° N/m^, Poisson’s ratio V12 = 0.25 and density p = 1.0x10
kg/m^. For comparison to results available in the literature one quarter of the
1451
shell is modeled by the proposed hybrid strain based shell element (identified
as HLCTS for brevity and convenience) with 4 x 4 D mesh. The boundary
conditions applied to the finite element model are: V = = 0.0 at line
AB,V = W = ©, = 0.0atBC,U = W = ©y = 0.0 at DC and U = ©y = ©, = 0.0
at AD. The number of equations to be solved after the application of the
boundary conditions is 189. A distributed step pressure is applied to its outer
surface (pointing toward the outer surface). It has an intensity p = 2000.0
N/ml The time step used were 0.03 s, 0.01 s and 0.005 s. As there was no
significant difference and for efficient reason throughout the computation the
time step of 0.03 s was adopted. The nonlinear transient response at the apex
(central point A of the shell) is obtained and plotted in Figure 4. The problem
has been solved by Wu et al. [14] who applied a curved high-order
quadrilateral shell element. The latter has 48 dof and was developed based on
the classical lamination theory. It is observed that there is a discrepancy of
about 8%, with respect to the HLCTS element results, for the amplitudes
between the two set of results. However, they have the same vibration period.
It is believed that the present results are more accurate as the element used in
the present investigation is shear deformable.
4.3 Cantilever Panel With Free End Step Moment
To demonstrate the use of the proposed shell element for structures
undergoing large rotation and large deformation a four layer cross-ply
cantilever panel is considered here. More computed results for this case can
be found in references [3] and [12]. It is symmetrically laminated with the
stacking scheme (0/90/90/0). Its geometrical properties are: L = 1.2 m, b = 0.1
m and h = 0.01 m. The material used for this cantilever is the high modulus
graphite/epoxy composite. Its properties are: Ej = 2.0685x10" N/m^, E2 =
5.1713x10'' N/m^ G,-, = 3.1028x10^ N/m^ G,, = G23 = 2.5856x10^ N/m^ p =
1605 kg/m^ and Poisson’s ratio Vj2 = 0.25. A step moment M about an axis
parallel to the width of the panel is applied to the free end. The amplitude of
this moment is = 1000.00 N-m.
The panel is discretized by a 12 x 1 A mesh. At the fixed end, all dof
are constrained. The finite element model has 144 unknowns.
The time step At = 0.001 s is employed in the trapezoidal rule direct
integration. The nonlinear transient responses at the end of the cantilever are
solved by selecting the options of director included, small strain and constant
thickness in the digital computer program developed. The computed end
deflections are plotted in Figure 5. As noted in reference [3,12], the inclusion
of directors in the formulation [15] is crucial as the directors are important
parameters that constitute the so-called "exact geometry" for large rotation
problems.
1452
5. CONCLUDING REMARKS
The hybrid strain based laminated composite flat triangular shell
(HLCTS) element for the static analysis of geometrically nonlinear laminated
composite plates and shells has been further developed and employed to solve
various dynamic problems. A relatively comprehensive study for various plate
and shell structures idealized by this element has been performed and three
representative examples are included to demonstrate its accuracy, efficiency
and conceptual adequacy. It is concluded that the HLCTS element is attractive
for large scale finite element analysis and modelling of shell structures
undergoing geometrically large deformation of finite strain and finite rotations.
ACKNOWLEDGMENT
The first author gratefully acknowledges the financial support in the
form of a research grant from the Natural Sciences and Engineering Research
Council of Canada. The results reported above were obtained in the course of
the research while the authors were at the University of Western Ontario.
REFERENCES
1. To, C.W.S. and Wang, B., Nonlinear theory and incremental
formulation of hybrid strain based composite laminated shell finite elements.
Proc. Second Int. Conf. on Composites Engineering, August 21-24, 1995, New
Orleans, Louisiana, pp. 757-758.
2. Wang, B. and To, C.W.S. , Finite element analysis of geometrically
nonlinear composite laminated plates and shells. Proc. Second Int. Conf. on
Composites Eng., August 21-24, 1995, New Orleans, Louisiana, pp. 791-792.
3. To, C.W.S. and Wang, Hybrid strain based geometrically nonlinear
laminated composite triangular shell elements, Part II: Numerical studies.
Comp, and Struct. (Submitted), 1996.
4. Liu, M.L. and To, C.W.S., Hybrid strain based three node flat
triangular shell elements. Part I: Nonlinear theory and incremental formulation.
Comput. Struct., 1995, 54, 1031-1056.
5. Noor, A.K. and Mathers, M.D., Nonlinear finite element analysis
of laminated composite shells. In Computational Methods in Nonlinear
Mechanics (Ed. by J.T. Oden, E.B. Becker, R.R. Craig, R.S. Dunham, C.P.
Johnson and W.L. Oberkampf). Proc. Int. Conf on Comput. Methods in
Nonlinear Mechanics, Austin, TX, 1974.
6. Lin, J.J., Fafard, M., Beaulieu, D. and Massicotte, B., Nonlinear
analysis of composite bridges by the finite element method. Comput. Struct.,
1991,40. 1151-1167.
1453
7. Madenci, E. and Bamt, A., A Free-formulation-based flat shell
element for nonlinear analysis of thin composite structures. Int. J. Numer.
Meth. Engng., 1994, 37, 3825-3842.
8. Bergan, P.G., and Nygard, M.K., Nonlinear shell analysis using free
formulation finite elements. In Finite Element Methods for Nonlinear
Problems, (edited by Bergan, P.G., Bathe, K.J., and Wunderlich, W.) Springer-
Verlag, 1986.
9. Zhu, J., Application of natural approach to nonlinear analysis of
sandwich and composite plates and shells. Comput. Meth. Appl Mech. Engng.,
1995, 120, 355-388.
10. Saleeb, A.F., Chang, T.Y., Graf, W. and Yingyeunyong, S., A
hybrid/mixed model for nonlinear shell analysis and its applications to large-
rotation problems, Int. J. Num.. Meth. Engng., 1990 29, 407-446.
11. To, C.W.S. and Wang, B., Hybrid strain-based three-node flat
triangular laminated composite shell elements for vibration analysis. J. Sound
and Vibration (submitted), 1996.
12. To, C.W.S. and Wang, B., Transient response analysis of
geometrically nonlinear laminated composite shell structures. Proc. of Design
Eng. Conf. and Computers in Eng. Conf. (edited by McCarthy, J.M.), August
18-22, 1996, Irvine, California, 96-DETC/CIE-1623.
13. Reddy, J.N., Geometrically nonlinear transient analysis of
laminated composite plates. A.I.A.A. J., 1987, 21, 621-629.
14. Wu, C.Y., Yang, T.Y. and Saigal, S., Free and forced nonlinear
dynamics of composite shell structures. J. Comp. Mat, 1987, 21, 898-909.
15. To, C.W.S. and Wang, B., Hybrid strain based geometrically
nonlinear laminated composite triangular shell elements, Part I: Theory and
element matrices. Computers and Structures (submitted), 1996.
A mesh
D mesh
Figure 1 Flat triangular laminated composite shell element
1454
Central deflection lY
Figure 2 Response of a cross-ply plate
Figure 3 Spherical shell segment under a uniformly distributed load
1455
Central deflection -W
Time b
Figure 4 Apex response with quarter shell considered
-0,2 0.0 0.2 0.4 0.6 0.0 1.0 1.2 1.4
X (m)
Figure 5 Evolution of cantilever panel with free end step moment
1456
ANALYTICAL METHODS III
THE FREE, IN-PLANE VIBRATION OF CIRCULAR RINGS
WITH SMALL THICKNESS VARIATIONS
R S Hwang, C H J Fox and S McWilliam
Department of Mechanical Engineering , University of Nottingham,
University Park, Nottingham NG7 2RD, England
Abstract
Geometric imperfections which cause thickness variations will always
exist in nominally circular rings and cylinders due to limitations in
manufacturing processes. The effects of circumferential thickness variations on
the natural frequencies of in-plane vibration are studied. The circumferential
variations in the inner and outer surfaces are describedj in a very general wayj
by means of Fourier series. Novozhilov thin-shell theory is used in conjunction
with the Rayleigh-Ritz method to obtain the natural frequencies. Results are
presented which show the effects of single-harmonic variations in the inner and
outer surface profiles, taking account of the profile amplitude of, and the spatial
phasing between, the inner and outer profiles. The frequency factors calculated
from the numerical method are in good agreement with those obtained from the
Finite Element method.
1. Introduction
The free vibrations of circular rings or shells had been studied by many
authors for over a century. The early theoretical works are summarised by
Love [I]. Most of these works are restricted to perfect rings or shells.
However, in practice, geometric imperfections (thickness variations and
departure from true circularity) are produced in the manufacturing process.
These affect the natural frequencies and mode shapes. It is weE known that in any
truly axisymmetric structure the vibration modes occur in degenerate pairs which
have equal natural frequencies and mode shapes which are spatially orthogonal but
of indeterminate circumferential location. The main effects of thickness variations
are to split the previous equal natural frequencies and remove the positional
indeterminacy [2]. Although these effects are often practically unimportant, there
are some applications (especially inertial sensors based on vibration rings or
cylinders [3] ) where the small frequency splits and fixing of the modal positions is
of primary practical significance. There is therefore a requirement to be able to
predict in detail the effects on vibrational behaviour due to small departures from
perfect circularity of the kind produced by manufacturing tolerances.
1457
The vibration of imperfect bells and rings were studied m the general
way using group theory [4,5]. In reference [5] the selection rules for frequency
splitting of thin circular rings were presented qualitatively. In reference[6] the
frequency splitting behaviour of a thin circular ring was investigated both
experimentally and analytically by first order perturbation theory. In reference
[7], the classical frequency equations, which are generally used to predict the
natural frequencies of a thin circular ring, were modified to describe an
eccentric ring by using the perturbation method. In reference [8], Fourier series
functions were used to represent the circumferential thickness variations of an
eccentric cylinder. Love thin-sheU theory, which is only strictly suitable for a
perfect ring or cylinder, was applied to investigate the free vibration of non-
circular shells.
In this paper, the free in-plane vibrations of thin rings of rectangular
cross section with circumferential variations in thickness are studied. The
circumferential variations in the inner and outer surfaces are described, in a very
general way, by means of Fourier series. Novozhilov thin-shell theory [9], in
conjunction with the well-known Rayleigh-Ritz method, are applied to analyse
the vibration characteristics for in-plane flexural vibration of the ring which is
considered as a special case of a thin shell [2,6]. The numerical method is used
to investigate the effect of single-harmonic circumferential variations in the
inner and outer surface profiles. The effects of harmonic number, amplitude and
spatial phasing between the inner and outer profiles are investigated. Some
important trends and patterns of effects of profile variations on the splitting of
the natural frequencies are observed. The results obtained by using the
numerical method developed in the current investigation are validated by
comparison with Finite Element predictions.
2. Method of Analysis
2.1 Geometry
Consider a thin ring of mean radius having a rectangular cross-
section of mean thickness h (« r«) and axial length L (« Ta). The inner and
outer surface vaiy along the global circumferential direction (Figure 1). rp
denotes the distance from the centre of the mean radius of the ring to the point
F on the middle surface.
Two coordinate systems are used in the formulation of the equation of
deformation.
1458
* Global polar coordinates (a', p', These are dir^ted along the global
axial, circumferential and radial directions. The initial geometry of the
undeformed, imperfect ring is defined using this coordinate system.
* Local curvilinear coordinates (a, p, These are directed along the local
axial, tangential and normal directions relative to the true middle surface and
coincident with the principal coordinates of the middle surface. This local
coordinate system is required for implementation of Novozhilov shell theory
which specifies displacements in the local tangential and normal directions.
is the angle between the global and local coordinate systems at the point P
of the middle surface.
All the displacements, thicknesses, and radii in this paper are expressed
dimensionlessly by dividing by lo , where k is the representative length and is
defmed as the mean radius of the ring.
Figure 1. A thin ring having circumferentially arbitrary surfaces
The shape of the middle surface of the ring is determined by the inner
and outer surfaces which can be expressed by Fourier series as follows:
f*(?')=U+'^f*cos(ip) + '^f; sin(jp) (1)
i=I j=I
/TP’)=/«'+ X/r«osc«P'-)+X/7««o‘P') (2)
i=/
1459
where / and / "(P') denote respectively the outer and inner surface
functions with respect to the global circumferential coordinate P and fo ,// j
//, fo', fi' and// are the Fourier coefficients which are defined in the usual
way [10].
The middle surface of a shell or ring is defined as the locus of the points
which lie at equal distances, * and from the outer and inner surfaces along
the direction normal to the mid surface (see Figure 2).
.. ^
Figure 2. The bounding surfaces and the middle surface
For given inner and outer surfaces, / (PpO and/'^(p/»0 defined in the
global coordinate system, the true middle surface can be determined using an
iterative numerical procedure which is fully described in [1 1]. Once the point P
on the true middle surface has been determined, the corresponding , Rp,
Tp and Pp can be calculated- These will be used in the step-by step integrations
which determine the strain energy and kinetic energy of the ring.
2.2 Equations of Motion
The strain energy for a thin ring whose length is much smaller than the
mean radius takes the form [2,6]:
S = «(!->■ l/R)d^d^ (3)
1460
Based on Novozhilov thin-shell theory, the normal strain epp in equation
(3) is given as
" 7 t / T> ^
(4)
1 + ^/ R
Ep= Vj^/R + w/R
(5)
Kp= - 1/R ( IV, p/j? +v/R), p
(6)
where Ep, Kp characterise the deformation of the middle surface of the thin ring
and subscript “ , p “ denotes partial derivatives with respect to (5. Epis the strain
tangential to the middle surface and Kp is the change of curvature, v, w are the
nondimensional local displacement components of the point P on the middle
surface along the tangential and normal directions respectively.
Substituting equations (4)-(6) into equation (3), then integrating with
respect to the thickness from hi~ to neglecting the 4th and higher powers
of /r F ^ and ^ f and noting that F(p) [ - hi~^] d?t = 0 where Ffp) is
an arbitrary function of p, the strain energy of a thin ring can be derived in
terms of the local displacements v as follows:
S= {[( V, p/ + 2h'v, p + w^]( 1 /R)[hi* -hi~ ]
2 rp;
jK
+ [ 2WW, p + 2w, p IV, pp - 2v w - 2viv, pp ] -h
3K
+ [(w,^)^ + v‘.2vw,f]^ ((l/R),f)^[kC^ -hr^]}d^ (7)
jK
Similarly, the kinetic energy of a thin ring, based on Novozhilov thin-
shell shell theory, can be expressed as follows:
T= {[(y,,)\(y^,,f]R[hC +
2 '•Pi
[3(v,t f- 4v,tW,,p + ( IV, /p f f (8)
in which p is the density of the ring, and the subscript ‘ ,/ * denotes the partial
derivatives with respect to time.
1461
For free vibration the tangential displacement v and the normal
displacement w 'which satisfy the boundary condition can be assumed to take the
following forms reg)ectively:
V = sinnp-v^ cosnp)^^^^ (9)
n
w = cosnp + w^ sinnp)^^^^ (10)
n
where v„ and are the undetermined amplitude coefficients of the tangential
and normal displacements of the middle surface respectively. The superscripts
"s" and "c” refer to the fact that these coefficients are multiplied by sine and
cosine terms respectively.
Substituting equations (9) and (10) into equations (7) and (8), then
applying the Rayleigh-Ritz procedure, the general frequency equation of the
free ■vibrations for a thin ring is obtained and can be expressed in the following
general matrix form:
-A
'M"
Ui~
~0~
0
where jfi: and M represent stiffiiess and mass matrices of size 2(N+1), and q
denotes a vector of generalised coordinates v„, w„ etc. The matrix elements in
equation (11) are given in [11]. Since in the general case the cross-section of
the ring 'wdl not be symmetric with respect to P = 0, the classification of the
modes as being “symmetric” and “antisymmetric” is meaningless. In the special
case of a perfect circular ring, the off-diagonal terms [K^J, /M"/ and
/Af7 appearing in equation (11) are null matrices, then equation (11) can be
uncoupled into two equations: one is for the symmetric modes and the other is
for the anti- symmetric modes -with respect to P=0.
The frequency factors of the ring. An , are the eigenvalues of equation
(11) and are defined by
A
0)1 d P
E
(12)
where a)„ is the natural frequency of the nth radial mode. The frequency factor
An is proportional to the square of natural frequency .
For a given value of n equation (11) will yield a pair of values of An.
These will be equal in the case of a perfect ring but will be slightly different in
1462
the case of an imperfect ring, giving rise to a higher frequency mode and a
lower frequency mode for each value of
It should be noted that the matrix elements in equation (11) are
expressed as integrals of the functions , Rp, yp and ^p with respect to
the tangential coordinate These functions are expressed in terms of the local
coordinates. Hence it is necessary to make a transformation to express these
functions and integrals in terms of the global coordinates, so that the integrals
can be evaluated over the global circumferential coordinate p* from 0 to 27t.
3. Results and Discussion
By using different combinations of trigonometric functions in equations
(1) and (2), it is in principle possible to model any closed thin ring. For the
purpose of illustration we will consider a nominally circular ring with a single
harmonic variation in the inner and outer surfaces, given by
(P') ~
/" OT = + V (JP’ - <!>)
where h/ and hf are the ampHtudes of the unperfections of the outer surface
and the inner surface measured from the mean outer radius ra and the mean
inner radius r/ respectively, is the spatial phase angle between the
trigonometric functions of the inner and outer surfaces at p* =0 , and /, j are the
harmonic numbers of the surface variations. Figure 3 illustrates i =] = 3 for
three values of ^ .
(^ = 0
<j> = 7t/2
^ %
Figure 3. Different spatial phase angles ^ for i = j = 3
1463
Results for the combinations of the geometric imperfections of i -J - 2,
3, 4, 5, 6, hf* - hf = O.lhy OMlh , and <J» = 0, 7c/4, nil, SjtM, tz are presented
here. The ring dimensions and material properties are as follows: =
40.75mm, r/ = 37.83mm, L = 2 ram = 206.7x10’ NW, p = 7850 kg/ml
Note that hf = O.lh corresponds to a departure from circular which is
much larger than would occur in practice due to imperfection. The results for hf —
OJh are presented to highlight the effects. Practically however, hf = OJDlh
represents a more realistic variation in thickness.
Convergence studies indicated that for hf* , hf~ = O.lh the use of 30 terms
in the solution series (equations 9 and 10) gave 4 significant figure accuracy or
better for the frequency factors A* fox k - 0, 1, ... 6. This was considered to be
acceptable for the purposes of the illustrative examples considered here.
In a parallel Finite Element study, beam elements, and two- and three-
dimensional plane stress elements used to model an imperfect ring. In order to
get 4 significant figures or better, 120 elements were used to model the
complete ring. Comparison of the results obtained from the numerical method
and the Finite Element Method shows that
(i) there is good agreement between the curves of frequency factors
obtained by the Finite Element method and the numerical method.
(ii) the trends and patterns of frequency splitting are nearly identical
irrespective of the analysis methods or the types of finite elements used.
In considering the effect of single harmonic variations of the profile of the
inner and outer surfaces on the natural fiequencies of different radial modes, the
discussion will focus on three aspects:
(a) , the effect of the harmonic number of the profile;
(b) . the effect of the magnitude of the profile variations; and
(c) . the effect of the spatial phasing between the profile variations of the
inner and outer surfaces.
The frequency splits shown are often very small (~ 0.001%). Note
however that in some inertial sensor applications, such small frequency splits
may be of practical significance.
1464
(a) The effect of profile harmonic number
Table 1 compares the frequency factors A„ obtained for a perfect ring
and an imperfect ring for ^-0, "^12, tc, h/" -hj — 0,lh, and i=j = 2 to 6. It is
evident from these results that:
For the flexural modes {n>2):
(i) When i , j are equal and even (see Table 1), frequency splitting only
occurs in the nth mode where w = Id. I 2 and /c is an integer. The
maximum frequency splitting occurs in the n—H 2 modes (i.e. k ^ 1 )
and the splitting decays as k increases.
(ii) When i , j are equal and odd (see Table 1) frequency splitting only
occurs in the nth mode where n = ki and k ism integer. The maximum
frequency splitting occurs for A: = 1, and splitting decreases as k
increases.
It should be noted here that frequency splitting in the higher modes
exists but is very small, e.g., for <}» = 0, i - j = 2 and kf= OJh , frequency
splitting occurs in the 2nd and higher radial modes. It can be seen from Table 1
that the splits in frequency factor are 0.019% at the 2nd mode, 0.001% at the
3rd mode, and less than 0.001% at the 4th mode or higher mode. These
correspond to actual frequency splits of about 0.01%, 0.(XK)5% and less than
0.CKX)5% respectively (equation (12)).
For the radial extensional mode( « = 0 ), no frequency splitting occurs.
It is clear from Table 1 that the trends and patterns of frequency splitting
are the same for ^-0,n/2 and tc. However, frequency splitting is less foT^ = 0
than for <}) = 71 under the same conditions. Frequency splits for ^ between 0 and
K are intermediate between those for <{» =0 and <{) = 7t .
The above patterns are in agreement with the qualitative results
published in reference [5] in which only the conditions for non-splitting are
established.
1465
Table 1
The difference of frequency factors A on the radial
modes w(n) [the parameters of profile variations are
taken as hf=0.1h/ i=j=2 to $ and (a)«|>=0; (b)i^5=TC/2; (c)4>=?c]
(a) 4>=
=0
A(0)
A(2)
A{3)
A(4)
A(5)
A(6)
Perfect
1.0005
0.003313
0.02645
0.09693
0.2525
0.5406
i= 2
high
low
-0.011%
-0.023%
-0.042%
-0.041%
-0.042%
-0,038%
-0.036%
-0.035%
i= 3
high
low
0.075%
-0.047%
-0.088%
-0.153%
-0.133%
-0.108%
-0.094%
-0.095%
i= 4
high
low
0.457%
2.155%
-2,287%
-0.134%
-0.150%
-0.594%
-0.349%
-0.240%
-0.248%
i= 5
high
low
1.631%
-0.171%
-0.141%
-0.332%
-0.229%
-1.816%
-0.838%
i= 6
high
low
5.656%
-0.171%
2.406%
-2.726%
-0.267%
-0.715%
-0.302%
-5’.624%
(b) 4)=Tr/2
A(0)
A(2)
A{3)
A(4)
A{5)
A(6)
i=
2
high
low
0.007%
0.336%
-2.427%
-0.595%
-0.676%
-0.971%
-0.972%
-1.169%
-1.272%
i=
3
high
low
0.090%
-1.231%
0.336%
-2.559%
-0.721%
-0.869%
-1.038%
-1.040%
i=
4
high
low
0.382%
14.58%
-18.71%
-1.150%
0.287%
-2.871%
-0.868%
-0.828%
-0.901%
i =
5
high
low
1.236%
-5.642%
-1.239%
-1.201%
0.212%
-3.703%
-1.172%
i =
6
high
low
4.111%
-5.620%
13.12%
-17.59%
-1.316%
-1.404%
0.117%
-6.370%
(c) 41=71
i =
2
high
low
A{0)
0.026%
A(2)
0.690%
-4.827%
A(3)
-1.132%
-1.340%
A(4)
-1,908%
-1.912%
A(5)
-2.308%
A{6)
-2.514%
i =
3
high
low
0.106%
-2.469%
0.709%
-4.934%
-1.312%
-1.637%
-1.989%
-1.993%
i=
4
high
low
0.314%
18.81%
-26.96%
-2.227%
0.528%
-4.993%
-1.388%
-1.447%
-1.532%
i=
5
high
low
0.881%
-10.89%
-2.400%
-2.135%
-0,018%
-5.031%
-1.510%
i=
6
high
low
2.808%
-10.81%
16.78%
-25.36%
-2.396%
-2.134%
-1.881%
-5.060%
Note:
1. difference = [A (n) - A (n)porfc.ct]x 100% / A (n) perfect
i\o
2. A(n) = — — (H (n) , where (ii(n) is the natural frequency at
E
the nth radial mode.
1466
Table 2. The difference of frequency factors A on the radial
inodes w(n) [the parameters of profile variations are
taken as hf=0.01h, i=j=2 to 6 and (a)<|>=0; (b)<|)=TC/2; (c)<|)=7t ]
(a) (|)=0
Perfect
A(0)
1.0005
A(2)
0.003313
A(3)
0.02645
A{4)
0.09693
A(5)
0.2525
A(6)
0.5406
i= 2 high
low
=0
=0
=0
=0
=0
=0
i= 3 high
low
0.001%
«0
-0.001%
-0.002%
-0.001%
-0.001%
-0.001%
i= 4 high
low
0.005%
0.222%
-0.223%
-0.001%
-0.001%
-0.006%
-0.004%
-0.002%
i= 5 high
low
0.016%
-0 . 002%
-0.002%
-0.003%
-0 . 002%
-0.019%
-0.009%
i= 6 high
low
0.060%
-0.002%
0.256%
-0.260%
-0.003%
-0.008%
-0.003%
-0-. 063%
(b) <j)=7t/2
A{0)
A(2)
A(3)
A(4)
A(5)
A(6)
i= 2
high
low
=0
0.004%
-0.024%
-0.006%
-0.010%
-0 . 012%
-0.013%
i= 3
high
low
0.001%
-0.012%
0.003%
-0.026%
-0.007%
-0.009%
-0.010%
i= 4
high
low
0.004%
1.684%
-1.725%
-0.011%
0.003%
-0 . 029%
-0 . 009%
-0.009%
i= 5
high
low
0.012%
-0 . 058%
-0 . 012%
-0.012%
0.002%
-0.038%
-0.012%
i= 6
high
low
0.044%
-0.057%
1.550%
-1.595%
-0 . 013%
-0.014%
0.001%
-0.069%
(C) <j>=7t
i= 2
high
low
A{0)
=0
A(2)
0 . 008%
-0 . 048%
A(3)
-0.012%
A(4)
-0.019%
A(5)
-0.023%
A(6)
-0.025%
i= 3
high
low
0.001%
-0 . 024%
0.008%
-0.050%
-0 . 013%
-0.016%
-0.020%
i= 4
high
low
0.003%
2.360%
-2.440%
-0.021%
0.006%
-0 . 050%
-0.014%
-0.015%
i= 5
high
low
0.009%
-0.114%
-0.023%
-0.020%
«0
-0.051%
-0.015%
i= 6
high
low
0.028%
-0.112%
2.165%
-2.251%
-0.023%
-0.020%
-0 . 018%
-0 . 051%
Note: 1. difference = [A (n) - A (n)p*r£ecc]x lOO^s / A(n}p<,r£«ot
2. A(n) = (£t^(n) . where tOfn) is the natural frequency of
E
the nth radial mode.
1467
(b) The elBfect of proBle amplitude
The effects of varied profile amplitude (h/' = hf = O.lh and OMlh) upon
the frequency factors for ^ = 0, it and i =J = 2 to 6 can be seen by
comparing Tables 1 and 2. It may be concluded from these results and others
which are presented in [1 1] that:
(1) When n-i 1 2, frequency factor splitting due to variable profile magnitude
compared with the frequency factor of the perfect ring is nearly
proportional to the profile amplitudes, hj* and hj .
For example, for i ^ = 7t and hf^ = hf 0,lh ,0.01h (see Tables 1
and 2) , the magnitude of frequency splitting at the 2nd mode is 45.77% for
hf = OJh and 4.80% for hf= OMlh. These correspond to actual frequency
splits of 24% and 2.4% respectively (equation (12))
(2) For modes other than those for which n^i 12, splitting of frequency factors
is nearly proportional to the square of the profile amplitudes, and hf .
For example, for i “4 , 2nd hf* = hf — O.lh ,0Mlh (see Tables 1
and 2), the magnitude of frequency factor splitting at the 4th mode are
5.521% for hf- O.lh and 0.056% for A/= O.Olh.
These results shown in Tables 1 and 2 for (j) = 0 and = ^t/2 show that
the general nature of the trends regarding the effect of profile amplitude
variations on the frequency factors are the same for all values of ^ , although
the magnitudes of the changes in frequency factors depend on ({>, as discussed in
the following section.
(c) The effect of spatial phase angle variations
The effects of the variations of spatial phase angle <j) on the frequency
factors are shown in Figure 4, from which it is evident that
(1) As frequency splitting occurs (see Figure 4.a-4.d ), the maximum frequency
splitting is obtained at <) = tc and the minimum splitting occurs at (p = 0. It
is clear that the maximum frequency splitting occurs in the n — il2 modes.
(2) In modes for which no frequency splitting occurs (see Figure 4.e and 4.f ),
the minimum frequency difference compared with that of the perfect ring is
detected at <j) = 0 and the maximum at ^ = n. Irrespective of the value of
(j) , the frequencies of these modes are always less than the corresponding
frequencies of the perfect ring.
1468
WKemnceIn 4 (%) at(eiwic»ln A (%)
Figure 4. Effect of spatial phasing on frequency factors with hf = O.lh and
(a) i=j=:4, 2nd mode ; (b) i=:j=6, 3rd mode
(c) i=j=2, 2nd mode ; (d) i=j=3, 3rd mode
(e) i=j=3, 4th mode ; (f) i=j=4, 5th mode
1469
4, Conclusions
In this paper, Novozhilov thin-shell theory and the Rayleigh-Ritz procedure
have been applied to derive the frequency equations of a thin ring with a rectangular
cross-section and a circumferential proffle variatioa Profile variations are
represented, in the general way, by Fourier series functions and the method gives
quantitative predictions of frequency splitting. The observed firequency splitting
patterns are in agreement with previously published qualitative results. Numerical
results have been presented for example cases in which the inner and outer profiles
are nominally circular with superimposed single-harmonic variations m radius. The
effects on frequency splitting of the harmonic number of the profile variation, and
the amplitude and spatial phasing between the inner and outer surfaces have been
investigated.
References
1. Love, A-E.H., A Treatise on the Mathematical Theory of Elasticity^ Dover
Publications, New York, fourth edition, 1952.
2. Fox, A simple theory for the analysis and correction of frequency
splitting in slightly imperfection rings. Journal of Sound and Vibration,
1990, 142(2), 227-43.
3- Fox, Vibrating cylinder rate gyro: theory of operation and error
analysis. Proceeding of DGON Symposium on Gyro Technology,
Stuttgart, 1988, Chapter 5.
4. Chamley, T. and Perrin, R., Studies with an eccentric bell. Journal of Sound
and Vibration, 1978, 58(4), 517-25.
5. Perrin, R., Selection rules for the splitting of the degenerate pairs of natural
frequencies of thin circular rings. Acustica, 1971, 25, 69-72.
6. Chamley, T. and Perrin, R., Perturbation studies with a thin circular ring.
Acustica, 1973, 28, 139-46.
7. Valkering, T.P. and Chamley, T., Radial vibrations of eccentric rings.
Journal of Sound and Vibration, 1983, 86(3), 369-93.
8. Tonin, R.F. and Bies, D.A., Free vibration of circular cylinders of variation
thickness. Journal of Sound and Vibration, 1979, 62(2), 165-80.
9. Novozhilov, V.V., The Theory of Thin Shells, P. Noordhoff Ltd., The
Netherlands, 1959.
10. Kreyszig, E., Advanced Engineering Mathematics, John Wiley & Sons,
Inc., Singapore, 1993, pp. 569-71.
11. Hwang, R., Free vibrations of a thin ring having circumferential profile
variations. Ph.D. Thesis, University of Nottingham, U,K, (in preparation)
1470
FREE VIBRATION ANALYSIS OF TRANSVERSE-SHEAR
DEFORMABLE RECTANGULAR PLATES
RESTING ON UNIFORM LATERAL ELASTIC EDGE SUPPORT
D.J. Gorman
University of Ottawa
770 King Edward Ave.,
Ottawa, Canada KIN 6N5
ABSTRACT
Utilizing the Superposition Method a free vibration analysis is conducted for
transverse-shear deformable rectangular plates resting on uniformly distributed
lateral elastic edge support. Edges are free of moment. The thick isotropic
Mindlin plate is utilized for illustrative purposes. The Mindlin equations are
satisfied throughout. Typical computed results are plotted for a square plate.
INTRODUCTION
It is well accepted that classical rectangular plate boundary conditions denoted
as simply supported or clamped are often not achieved in real structures. This is
because of elasticity in the edge supports. Furthermore, in many rectangular
plate installations elastic edge supports may be utilized intentionally. For this
reason a number of studies of effects of elasticity in the edge supports on
rectangular plate free vibration frequencies have been conducted and results
published. Almost all of these studies have been devoted to the free vibration
behaviour of thin isotropic plates. Studies by the author, related to this family of
vibration problems, have been devoted to situations where elastic stiffness is
uniformly distributed along the edges as well as cases where the stiffnesses are
arbitrarily distributed. All of his studies have been conducted by means or the
Superposition Method and in a fairly recent article he has demonstrated that all
of these families are amenable to analytical type solutions [1].
In this paper we exploit the powerful Superposition Method to analyse the free
vibration behaviour of transverse-shear deformable plates resting on uniform
lateral elastic edge support. This represents a much more complicated problem
than the thin isotropic plate problems discussed above. For our purposes we
choose the thick shear-deformable Mindlin plate and base our solution on
Mindlin theory.
In the interest of keeping the literature review up to date the recent publication
1471
of SAHA, KAR, and DATTA [2] is drawn to the attention of the reader. They
report on a study of thick Mindlin plates resting on edge supports with uniform
lateral and rotational elasticity. They have employed a Rayleigh-Rite energy
approach. Plate lateral displacement is represented by a rather complicated set
of Timoshenko beam functions, each extremity of each beam being attached to
a local lateral and torsional spring. It will be seen that no such functions need be
selected in the superposition approach adopted here. Another related paper is
one by the present author dealing with Mindlin plates where lateral displacement
along the plate edges is forbidden but uniform rotational elastic support is
provided [3]. This problem is somewhat easier to solve since edge lateral
displacement is forbidden and, unlike the present problem, mixed derivatives do
not show up in the boundary condition formulation. This latter problem was
shown to be amenable to solution by the modified Superposition-Galerkin
Method which is extremely easy to use when it is applicable.
MATHEMATICAL PROCEEDURE
A solution to the present problem is obtained through th& supei-position of the
eic^ht edge-driven forced vibration solutions (building blocks) shown
scheraaticaly in Figure 1. All of the non-driven edges have slip-shear support.
This type of support, indicated in the figure by two small circles adjacent to the
edc^e, implies that the edge is free of torsional moment and transverse shear
fomes. Furthermore, rotation of the plate cross-section along the edge is
everywhere zero.
We begin by examining the first building block. Its driven edge is free of
torsional moment, and rotation of the plate cross-section along this boundary is
every where zero. This latter condition is indicated by two solid dots adjacent to
the edge. Driving of this edge is accomplished by a distributed harmonic
transverse sheai* force of circular frequency o). The spatial distribution of the
shear force is expressed as,
Q,l,.,= E E„cos(m-l)7t5 (1)
m = 1 ,2
where k is the number of terms required in the series.
1472
Fig. 1 Schematic representation of building blocks utilised in theoretical
analysis.
We now examine the response of the above building block to this harmonic
excitation. The proceedure followed is almost identical to that described in an
earlier publication [4]. A concise description will be provided here for the sake
of completeness.
The governing differential equations which control the response of thick Mindlin
plates are written in dimensionless form as,
W d" W ^ ^ ^ W = 0 (2)
8^“ (j)- art“ ({) ari V3
d^- 4)- 8 Ti^ 4) 5 ^ 8 r| i ^ 5 U
(3)
8f 4)^v, a-n- 4)Vi d^dr\
1473
Transverse shear forces, bending moments, etc., are written as.
V 9 „
^ 3^ cj) 3r| ^ 3ti
(j) 3n
at a tf 9 <|i- 1 a i(r.
When subjected to the first term of the driving force (Eqn 1) the response
of the building block will be essentially that of a Timoshenko beam. The
governing differential equations reduce to a set of two which may be
written as
d“W ^ d-ijj A'^(})“c|);;W
- + (p - !• + -
d rt- d Ti"“ V3
= 0
(6)
and
d'l};^ V3<{)V ^ 1 dW K
^2 dn j 12
(7)
It is convenient to represent the lateral displacement W, and plate cross-
section rotation as,
W(Ti) = X(Ti), and (ti) = Z (t))
The governing differential equations may then be written as
X"(q) + a^,Z'(q) + b„,X(ti) = 0 (8)
and
Z''(q) + a,^3X'(n) + b„,5Z(Ti) = 0 (9)
where superscripts imply differentiation with respect to r|. Coefficients
1474
.... etc,, are defined in reference [4].
Applying the appropriate differential operators to this set of equations
the parameter X (ti) is eliminated and a second order ordinary
homogenous differential equation is obtained involving the parameter Z
(q). It is found that for our range of interest the roots of the characteristic
equation associated with this differential equation are always real. There
are then three possible pairs of roots depending on the coefficients in the
above differential equation. Designating these pairs of real roots as
Rj and possible forms of solution exist as follows,
Casel, RpR2<0-0 Case2, R,<0-0; R,>0-0 CaseS, Ri,R2>0-0 (lO)
In all work reported here it has been found that only, case 2, has been
encountered.
It will be obvious that the functions X (q) and Z (q) must be symmetric
with respect to the ^ axis. We may therefore write for case 2,
X(q) = cos aq + cosh pq, (11)
and
Z(^) = Am ( cos a q + cosh P q (12)
where a = ^|R, | , and P =
Expressions and S^.^re obtained by taking advantage of the
coupling of equations 8 and 9, as was done in Reference [4].
We then impose the boundary conditions, Q^ = E,^^,and ilf^ = 0,at q=l,
in order to evaluate the unknowns and B^^^ of Equation (11).
Accordingly we obtain,
E
X(q) = .^{cos aq + X 1 cosh Pq}, (13)
and
1475
Z(T1) = ^ {S„ , sin a n + XI S„, sinh P t|}, (14)
where XI and X2 are easily evaluated.
Next we examine the response of the first building block to driving terms
where m>l. We follow the proceedure described in Reference [4].
Levy type solutions for the parameters W, ....
. etc., are written as.
W(5,ti) = XjTi) cosirni?
(15)
litres, Tl) = Y„(ri) sinm-ii^
• (16)
= Z„(T|) cosmn?
(17)
It will be noted that all required boundary conditions along the edges, ^
= 0, and ^ = 1/ are satisfied.
Next, the above expressions (Eqns 15, 16, 17) are substituted in the set of
governing differential equations. The following set of coupled ordinary
differential equations, written in matrix form are obtained
X
0
0
X./'
t>n,l
^m2
0
x„
o'
Y
^ m
0
0
^m2
Y.'
^m4
0
Y„,
► _ 4
0
zj
. m J
.^m3
^m4
0
0
0
bn,5.
0
(18)
Again, the quantities .... etc., are defined in Reference [4].
Applying the appropriate operators on the above equations, as was done
in Reference [4] we are able isolate a single homogenous sixth order
ordinary differential equation involving the dependent variable (ti),
only. Because first, third, and fifth order derivates are missing from this
equation the associated characteristic equation can be formulated as a
cubic algebraic equation. Again it is found that for the range of the
1476
present study all of the roots are real Designating these roots^ as
R ,R,, and R3, it follows that four solution cases are possible depending
orl the coefficients of the characteristic equation. They are.
Case!, R, , R,, and R. < 0-0 Case2, R, , R^ < 0-0 ; R3 > 0-0
‘ - (19)
Cases, Rj < 0-0 ; R, and R3 > 0-0 Case4, Rp R, and R3 > 0-0
Inthepresentstudyonlycase3,andcase4are€ncountered.Introdudng a=^\ Rj 1 ,
P = , a n d y = JIR3 1 and recognizing that (q ) must be
antisyrnmetric about the ^ axis while X^(q)and Y^(q)must be
symmetric, we are able to write for case 4,
Y^(q) = A,^ cosh aq+B^^ cosh Pq + C^^^ cosh yq
(20)
Utilizing the coupling of the ordinary differential equations, as in
Reference [4], it follows that we may write,
X^/q) = A^R,^i cosh ocq + B^R^, N + (21)
and
Z^(q) = A^S^, cosh ocq + B^^S^, cosh Pq+C,^iS^3 cosh yq (22)
The quantities R^, , , S^p ... etc., are evaluated following steps described
in Reference [4]. Expressions for (q ) , X^^ (q ) , etc., for case 3 will differ
from the above expressions only iii that Cosh a q must be replaced by cos
a q.
The unknown constants A^ , B „p etc., of the above solutions are evaluated
by enforcement of boundary conditions along the edge, q=l. These
conditions comprise zero torsional moment, zero edge rotation, with
transverse shear force = E^p For case 4 we obtain
(n) = — { cosh a q + X 1 cosh p q + X 2 cosh y q } (23)
with the functions X^(q) and (q ) differing from Y,^(q) only in that
1477
R^j and S^j , etc., must be included.
We therefore now have the exact response of the first building block to
the imposed driving force components available. It will be observed in
Figure 1 that the second, fifth, and sixth building blocks differ from the
first only in that they are driven along different edges. Solutions for their
response are therefore easily extracted from that of the first.
Focusing our attention on the third building block we find that its driven
edge is free of transverse shear forces and torsional moment. It is driven
by a distributed cross-section harmonic rotation. The spacial distribution
of this imposed driving rotation is also represented by the series of
Equation 1.
The reader will appreciate that a solution for the third building block is
obtained by following steps identical to those described for the first. Only
the imposed boimdary conditions along the driven edge differ. Solutions
for the quantities W, etc., will be identical in form to those already
developed for the first building block except that quantities XI, XI, etc.,
will be slightly different. We designate them as XIP, X2P, etc., for the
edge-rotation driven building blocks. Solutions for the remaining four
building blocks of Figure 1 are therefore available.
THE EIGENVALUE MATRIX
This matrix is shown schematicaly in Figure 2. It is generated following
established practices. Let us first consider the transverse force
equilibrium condition along the edge, rj = 1. It is readily shown that this
equilibrium condition is written in dimensionless form as,
Q, + K„W = 0 (24)
The plus sign of this equation must be replaced by a minus sign when we
formulate the corresponding equations for the edges, 11=0, and ^=0.
1478
Em
1 2 3
^ En
1 2 3
c:>
CM
C'> -
W4
^ Ep
i 2 3
- 1 10
^ E,
1 2 3
r&ean-,
% <
^ Es
1 2 3
1 2 3
-
-
- - ■
-
:::
-
: : :
LvJ
: : :
-
- - -
-
_ - -
-
: : :
-
-
« - -
-
Ill
-
: : :
-
1
1 M 1
t
p-
- ^ -
- - -
-
: : :
: : :
-
□
-
: : :
-
-
-
: : :
* _
[v
+
r
_ - -
-
- - -
-
- “ "
-
: : :
■
-
: : :
-
: : :
Ill
-
: : :
1 M
1
i"
j
]'
: : :
-
: : :
-
" " I
■
I I -
Pig 2 Schematic representation of Eigenvalue matrix based on
three-term function expansions. Short bars indicate non¬
zero elements. M or V on inserts to right indicate edges
along which moment or lateral force equilibrium is
enforced.
To construct the first three equations upon which this matrix is based we
superimpose all eight building blocks and expand their net contribution
to displacement W in a cosine series. The transverse shear force along the
edge, 11=1, is already available in such a series. We then express the left
hand side of equation 24 in series form and require that each net
coefficient in this series must vanish. This leads to 3 homogenous
algebraic equations relating the 8 k imknowns where, for the illustrative
matrix of Figure 2, k equals 3.
A second set of three homogenous algebraic equations is obtained by
enforcing the corresponding lateral equilibrium condition along the edge.
1479
^ = 1. Moving down the matrix of Figure 2 it is seen that a third and
fourth set of equations are obtained by enforcing the moment
equilibrium condition, i.e., net bending moment equals zero along the
same edges, in an identical fashion. Finally, it is seen in Figure 2 that four
more sets of equations are obtained by enforcing the required
equilibrium conditions along the edges, ti=0, and ^=0.
We thus have, in general, 8 k homogenous algebraic equations relating
the 8 k unknown driving coefficients. The coefficient matrix of this total
set of equations forms our Eigenvalue matrix.
Certain measures can be taken to greatly simplify and expedite
generation of the matrix. It will be observed (Fig. 2) that the matrix is
composed of 64 natural segments. This array of segments may be
referred to through the indices (I, J). It is expedient to first generate the
matrix without including contributions related to the driving shear
forces along the building block edges (Eqn. 24). The matrix is then
completed by adding the quantity 1.0 to diagonal elements of segments
(1,1) and (2,2), and subtracting 1.0 from the diagonal elements of
segments (5,5) and (6,6).
Physical reasoning leads also to another vast signification. One may
begin by generating the elements of the matrix lying below the first four
building blocks, only, (Fig. 2). Following a proceedure as discussed in
Reference [4], and exercising caution with respect to necessary sign
changes, all of the remaining segments of the matrix may be extracted
from those already generated.
Eigenvalues are, of course, those values of the dimensionless frequency,
X-, which cause the determinant of the Eigenvalue matrix to vanish.
Mode shapes are obtained after setting one of the non-zero driving
coefficients equal to zero and solving for the others.
PRESENTATION OF COMPUTED RESULTS
It will be appreciated that problems involving vast arrays of stiffness
coefficients, plate aspect ratios, thickness-to-length ratios, etc. can be
resolved by the proceedure described above. Only a single typical
problem and its solution will be discussed here, for illustrative purposes.
We consider a square plate with equal dimensionless lateral elastic
1480
stiffness imposed along each edge. Results are presented for two
thickness-to-length ratios, 0.01, and 0.1. Two important observations
may be made before examining these results. First, for the very thin plate
of thickness-to-length ratio, 0.01, we expect the Eigenvalue vs edge-
stiffness ratio curves to almost co-inside with those for a thin isotropic
plate based on thin plate theory. Secondly, we recall that the Eigenvalue
limits for a thin plate will equal 0.0, and 2 as the elastic stiffnesse
approaches natural limits of 0.0, and infinity.
Results of a free vibration study of the above plate are presented in
Figure 3. It will be noted that computed Eigenvalues are plotted against
the parameter Kli / (j)J. By presenting data in this manner it is found
1481
that Eigenvalues for both the thick and the thin plate can be plotted on
the same Figure. The absissa of the figure appears in five logarithmic
decades. This range has been selected with a view to providing
information for the reader over the region of greatest interest. It is found
that in fact the thin-plate curve approaches the known limits disucssed
above. Furthermore, utilizing computing schemes related to Reference
[1], it has been shown that this same curve lies extremely close to that
obtained for thin plates based on thin plate theory. Because Mindlin
theory takes rotary inertia into effect and does not consider resistance to
transverse-shear induced deformation to be infinite the thin plate curve
of Figure 3 will lie only perceptably below that of its companion curve
based on classical thin plate theory. The lower limit for the thicker plate
curve of Figure 3 will also equal zero. However, the upper limit, equal
to that of a simply supported thick Mindlin plate will lie below the
classical value of 2n^. This is seen to be the case in Figure 3.
DISCUSSION AND CONCLUSIONS
The superposition technique is seen to constitute an accurate, straight¬
forward technique for obtaining analytical type solutions to the problem
of analysing free vibration of Mindlin plates resting on uniform lateral
elastic edge support. Convergence is found to be rapid. The seven terms
utilized here in representing building block solutions are found to be
more than sufficient to provide four digit accuracy in computed
Eigenvalues.
The reader will appreciate that the analysis described here could easily
be modified to handle plates with lateral elastic support along less than
four edges. Plates, for example, with two adjacent free edges and the
other two given uniform lateral elastic edge support can easily be
handled. It is only necessary to set the elastic stiffness coefficients equal
to zero for the first two edges. It will also be appreciated that through
proper choice of building blocks one can analyse plates with one or more
edges resting on elastic support while the others are given various
combinations of simply supported, clamped, or free boimdary conditions.
Natural extensions of the present analysis would lead to the solution of
problems involving the same plates resting on arbitrarily distributed
elastic edge support and combinations of lateral and rotational elastic
support.
1482
LIST OF SYMBOLS
a, b
D
E
G
h
ki, k2, etc..
Kur Kl2, etc.,
Plate edge dimensions
- E hV(12 (1 - )), Plate flexural rigidity
Youngs modulus of plate material
Modulus of elasticity in shear of plate material
Plate thickness
Basic lateral spring stiffness along plate edge.
Subscript 1 indicates edge, ti=1. 2, 3, 4, indicate edges
moving counter-clockwise from 1.
Dimensionless lateral elastic edge coefficients,
k, aw k.,aw
_! _ - etc.
K^-Gh ’ K-Gh'
M^, M,
w
■y
K"
V
V2
V3
4)
4>h
Dimensionless bending moments associated with ^
and r\ directions, respectively
Dimensionless twisting moment
Dimensionless shear forces associated with ^ and r|
directions, respectively.
Plate lateral displacement divided by side length a
Distances along plate co-ordinate axes divided by
side lengths a, and b, respectively
Mindlin shear factor = 0.8601
Poisson ratio of plate material
= (l-v)/2
= (1 + v)/2
= 6 (1 - v)
Plate aspect ratio = b/a
Plate thickness ratio = h/a
1483
}\f^ Plate cross-section rotations associated with ^ and r\
directions, respectively
CO Circular frequency of plate vibration
= CO a “ ^p/D, Free Vibration Eigenvalue
p Mass of plate per unit area
REFERENCES
1) Gorman, DJ., “A General Solution for the Free Vibration of
Rectangular Plates with Arbitrarily Distributed Lateral and
Rotational Elastic Edge Support”, Journal of Sound and Vibration,
1994, Vol. (174), No. 4, 451-459.
2) Saha, K. N., Kar, R.C. , and Datta, P.K., “Free Vibration Analysis
of Rectangular Mindlin Plates with Elastic Restraints Uniformly
Distributed Along the Edges”, Journal of Sound and Vibration, 1996,
Vol. (192), No. 4, 885-904.
3) Gorman, D.J., “Accurate Free Vibration Analysis of Shear-
Deformable Plates with Torsional Elastic Edge Support”, to be
published in Journal of Sound & Vibration.
4) Gorman, D.J., “Accurate Free Vibration Analysis of the Completely
Free Rectangular Mindlin Plate”, Journal of Sound and Vibration,
1996, Vol. (189), No. 3, 341-353.
1484
Wave Equation Eigensolutions on Asymmetric Domains
R. G. Parker
Department of Mechanical Engineering
Ohio State University
C. D. Mote, Jr.
Vice Chancellor-University Relations, FANUC Chair in Mechanical Systems
Department of Mechanical Engineering
University of California, Berkeley
The wave equation
Introduction
- V-^ -i- =/(x,y,r) P
q+^qn^O dP
(1)
is arguably the most widely-studied differential equation in science and engineering. It is used to model
physical systems in diverse fields such as acoustics, wave propagation, vibration, electromagnetics, fluid
mechanics, heat transfer, and diffusion. Eigensolutions of the two-dimensional wave equation, governed
by the Helmholtz equation, are the focus of this paper. We present a perturbation method to analytically
calculate eigensolutions when asymmetric perturbations are present in the boundary conditioiis. The
axisymmetric, annular domain case serves asjthe unperturbed problem. Possible boundary condition p^-
turbations include deviation of the domain P from annular and variation of the parameter p along the
boundary dP . The eigensolution perturbations are determined exactly, and their algebraic simplicity
allows extension of the perturbation through fifth order. Both distinct and degenerate eigenvalues of the
unperturbed problem are examined. Boundary condition asymmetry splits the degenerate unperturbed
eigenvalues. We derive simple rules predicting this splitting at both first and second orders of perturbation.
To illustrate the method and quantify its accuracy, the case of domain shape perturbation from circular is
addressed in detail and comparisons are made with the exact solutions for elliptical and rectangular
domains. . .
Methods used to analyze eigensolutions of the wave equation on irregular domains have been pri¬
marily numerical; for example finite element, finite difference and others. In addition to an extensive sum¬
mary of theoretical results, Kuttler and Sigillito [1] provide a comprehensive review (142 references) of the
application of these and other less popular methods. Mazumdar also reviews approximate methods
invoked for this problem [2,3,4]. The above methods can be augmented by conformally mapping the
irregular domain to a circle [5]. In the spirit of perturbation, Joseph [6] employed a parameter differentia¬
tion method to obtain derivatives of the distinct eigenvalues as the domain changes. The requirement of a
smooth mapping function from the unperturbed domain to the irregular one and the restriction to distinct
eigenvalues of the unperturbed problem limit its applicability. By assuming expressions for the lines of
constant deflection in the fundamental eigenfunction, Mazumdar obtained estimates for the fundamental
eigenvalue for arbitrarily shaped domains [7]. Accuracy of this method depends on the availability of a
good estimate of the lines of constant deflection. Morse and Feshbach [8] used a perturbation analysis dif¬
ferent than that presented herein to study the Helmholtz equation on irregular domains. Expansion of the
eigenfunction perturbations in infinite series of the unperturbed eigenfunctions leads to a convergence
problem restricting the analysis to second order perturbation in the eigenvalue and first order in ^e eigen¬
function. Nayfeh used a perturbation formulation similar to that of this work to calculate the eigenvalue
perturbation to first order; no eigenfunction perturbations are presented [9]. This work draws on the results
of Parker and Mote [10, 11], where a formal procedure for obtaining exact eigensolution perturbations is
developed.
1485
Eigensolution Perturbation Formulation
The eigenvalue problem .resulting from separation of the spatial and temporal dependence in (1) by
the assumption q =/?(/?, 0)e is
- VV - = 0 P (2a)
p + = 0 : dPi [j dP(, (2b)
where subscript n denotes the normal derivative. We examine two classes of asymmetry that_normally
preclude exact determination of the exact solution to (2): irregular P and variation of P along dP . These
asymmetries are treated as perturbations of the axisymmetric, annular domain eigenvalue ^oblem.
(i) Irregular Domain Shape: The two-dimensional, doubly-connected domain P in Fig. 1 is
P: Ri(Q)<R <Ro(Q),0<Q< 2%. The deviations of the boundaries dPi and dPg from circular are
£^.(0)^, (0) ■ Ri (6) - Ro (3)
where Ri and R^ are the average radii of the inner and outer boundaries. The variables q,p, and t of (1)
and (2) are dimensionless; scaling the domain with the additional dimensionless variables
r = -§- Y=^ (a = R,a eg,(9)
Pq Po
Ego(Q)
Ro
-
Pq
(4)
yields the eigenvalue problem for constant P = Po
- V^p - (£>^p = 0
P
(5a)
p +^oPn=0
3?
(5b)
where?: Y-i- £g,(0)<r <1 + g^^(0), O<0<27t.
Boundary quantities on dP are approximated by Taylor series expansion about r = y, 1. For exam-
ple,
? lr=I+£g, = ? Ir=l + i^8o)Pr I r=l + Prr \ r=l + Prrr 1 r=I + ‘ ' (6)
A similar expansion is developed for p ] 3^, . The expansions | require asymptotic expansion of p,i
in terms of derivatiyes with respect to the polar coordinates r and ^ [12]. Introduction of the expansions
for p and p^ on dP into (5b) yields
- = 0 P: Y<r <1, O<0<27t (7a)
(? -Po?r) + eQ? + ■ ■ ■ =0 r=y (7b)
ip -f Po Pr) + zCp + Z^Dp +z^Ep + • • • = 0 r = l (7c)
where C ,D ,C ,D , ‘ • are linear boundary operators with variable coefficients depending on (0) and
8
(ii) Variable p along dP : For annular domains with P — > P^, + E p(0), the eigenvalue problem becomes
-W^p -a^p =0 ?:Y<r <l,O<0<27t (8a)
(? - Po?r)-£p(0)?r =(? - PoPr) + =0 ^=7 (8b)
(? + PoPr) + eP(0)?r =(? + PoPr) + = 0 '‘ = 1 (8c)
which is of form identical to (7).
We seek eigensolutions of (7) where the boundary perturbations may result from either or both of the
asymmetries discussed above. The eigensolutions are represented as asymptotic series in the small param¬
eter £
1486
03^ = 03^ + ep. + e^T| + + + O(s^)
(9)
p=u + EV + z^w +z^s +£^r +eh +0(£^) (10)
Subsequent analysis shows that confinement of the perturbation terms to the boundary conditions ensures
that the Bessel and trigonometric forms of the eigenfunction perturbations in (10) do not depend on the
boundary perturbations. Only the coefficients of these Bessel and trigonometric functions depend on the
boundary condition operators. Because of this essential point, the method presented in the sequel for
finding exact solutions for the eigensolution perturbations on irregular domains can be readily applied to
find exact eigensolution perturbations for any problem of the form (7).
With the inner product <e ,f > dA,\ht normalization <p ,p> = 1 and (10) give
P
<u,v> = 0 <u,w> = -V2<v,v> <u,s>=-<v,w>
<u,t> = ~ <v,s> - V2< w , w >
(11)
The eigenvalue perturbations li,T|, K,X, and G are determined subsequently in terms of the boundary
conjuncty (£ , / )
J(e,f) = < -Vhj>-<e. -VV>= j [e/„ ■fe„]ds- (12)
dP
Irregular Domain Eigensolution Perturbation
We demonstrate the solution procedure for-ei gen value problems of the form (7) by examining an
irregularly-shaped domain with outer boundary dP and boundary condition p 1 gp - 0. Use of (4) and (6)
yields
-V^p-'^^p=0 P:0<r <1,O<0<27C (13a)
p +Zgp,+ 1/lKtgfprr + \m^gfpnr + WK^gtPrrrr + l/5!(£« =0 r = 1 (13b)
where the subscript o denoting the outer boundary has been omitted. Substitution of (9, 10) into (13)
yields the sequence of perturbation problems
- M - 03^ M = 0 P
(14a)
U = 0 dP
(14b)
- V - 03^ V = jl W P
(15a)
V ^ -g Ur dP
(15b)
-V“W -03~W = |iV -I-TIM
P
(16a)
W = -gV, ~{m)g\r
dP
(16b)
-05^ 5 = |IW -fTlV -l-KW
P
(17a)
5 = - gw r - (1/2) g\rr * 8^^rrr
dP
(17b)
-V^r-03^r -\is +r\w + kv + X“
P
(18a)
t = -gSr - (l/2)g Vr - (l/6)5\rr "
{1/24) g\rrr
(18b)
1487
-V^z -CO^Z =|lf +T15 -1-KW +XV +GU
p
(19a)
z - - gt,- - (l/6)g Vrr - (l/24)g Vrrr
-(l/120)gVrrr
(19b)
Solution of (14) gives the orthonormal unperturbed eigenfunctions
1
^mO ~ lA T f r,\ \
m>0
(20)
cs 2 >^0 ( ®/nn ) [cos n0 d / ^\l
cos n 0 -..n A
sinne '«^0,n>0
(21)
where m and n denote the number of nodal circles and nodal diameters in the eigenfunction. The unper¬
turbed eigenvalue is the (m + 1)'" root of the characteristic equation (CO) = 0. C0^„ is a distinct
eigenvalue for « =0 and a degenerate eigenvalue of multiplicity two for n > 1. are orthonormal
eigenfunctions associated with the degenerate eigenvalues.
For the circular domain P , the boundary conjunct (12) is
f [efr -ferUidQ
0
The following relations are used subsequently
o' n'*-
“ ' ^i/j
K
Rmna)= -
r 1
2
V2
2
%
^mn ■^7««(1) ~
K
g ( ©) = X cosy 0 + X sin; 0
;=i 7=1
(22)
/z
C0^,i:23)
(24)
(25)
G(0) = g2(0) = Go + X G/cos;0 + X OjsinjQ
7=1 7=1
Use of the Fourier representation (24) allows treatment of arbitrary boundary shapes, including kinked or
discontinuous boundaries. The constant term in (24) vanishes because R is the average radius of the boun¬
dary.
Solution of Perturbation Equations
Distinct Eigenvalue Perturbation
Consider perturbation of a distinct unperturbed eigensolution (CO;„o> ^mo) (lii® subscript mO will be
omitted in the sequel). Using solvability conditions for the perturbation problems (15-19), Parker and Mote
present formal expressions for the distinct eigenvalue perturbations in terms of the boundary conjunct
[10,11]
p. = -J{u,v) Tl = -Jiu,w) K= -\i<u,w > - J(^u,s ) (26)
X= - \1<U,S> -T\<U,W> ‘ J(u,t) a= -\L<U,t> -r\<U,S> ~K<U,W> -J(u,z)
1st Order Perturbation: The first order eigenvalue perturbation }I is evaluated from (26a), (22), (14b),
(15b), (20), and (24)
2tc 2n
0 0
The result p. = 0 substantially simplifies subsequent calculations. It results because the radius of the unper¬
turbed circular domain is the mean radius of the irregular domain.
1488
In their treatment of boundary shape perturbation of the Helmholtz equation, Morse and Feshbach
[8] noted convergence difficulties when the eigenfunction perturbation is expanded in a series of the unper¬
turbed eigenfunctions. For the similar case of plate boundary shape perturbation, Parker and Mote [12]
encountered a divergent series for the second order eigenvalue perturbation when the first order eigenfunc¬
tion perturbation is expanded in a series of the unperturbed eigenfunctions. These problems do not occur
when the exact solution for the eigenfunction perturbation is determined. Furthemore, the exact solufion
is more accurate, computationally efficient, and notationally conveiiient than the infinite series expansion.
In the sequel, exact eigenfunction perturbation solutions are determined through fourth order perturbation,
thereby allowing exact calculation of the fifth order eigenvalue perturbation.
The eigenfunction perturbation v ( r , 9) is decomposed as
V = c « -h (28)
The first term results because w is a non-trivial solution of the homogeneous form of (15a,b); c is a con¬
stant to be determined. The second term is the general solution of the homogeneous form of (15a). The
third term of (28) is a particular solution of the inhomogeneous equation (15a), Because of (27),
= 0 (29)
Additionally,
= £y;(cor)[i5y cosjQ + Cj sinyG] (30)
where the j = 0 term is omitted because its contribution is included in the first term of (28). The
coefficients in (30) are calculated from (15b)
^ _ ®g/
rVy(CO)
= I/.
^ 7t^Jj(C0)
(31)
(32)
where and are from (24). Substitution of (28) into the normalization condition (11a) yields
c - - <M, v^' -f v^> = 0
and the solution for v is complete.
2nd Order Perturbation: The second order eigenvalue perturbation T| is evaluated from (26b), (22),
(14b), (16b), (20), (28-32), (24), and (25)
Tl = j [Mr ( - g^ - V2g\r)'\r^idQ = ^0+1. ^
The solution to (16) is decomposed as
w = d u +
where the definitions of the terms in (34) are analogous to those in (28), and
r\r d t(cor)
vP = -
(47t)‘'^C0 ^i(co)
(34)
(35)
(36)
w -'Z dj (wr ) [ Ej cos; 6 -i- Fj sin j 0 ]
j=i
The coefficients Ej and Fj in (36) are determined from (16b)
^ gm(gm+; 8m-j^ 8m^8m+J 8m~j^ ^ ^7
r-’C r dfji+ii(ii)
Gf+ X -ft) , . . 3
£,= -
(47:)'^Vy(0))
4 (CO)
8m^8m+j -^gy-m) g»i(gm+; "gy-m) ^
m =;
8m 8 2m 8m 8 2m
1489
Fj=.
CO
From the normalization (11b),
^ r T J
8m^8m+j ~ 8m-j^ ~ 8m^8m+j ~ 8m-j^
8m^8m-^i 8j-m) ~ 8m(8m+j ~ 8j~m)
[_ 8m 8 2m ~ 8m 8 2m
1-
d = -V2<v,v>-<u,wP>
’m^l (CO)
2m
■ +
CO ^(CO) y^(co)
m >j
m <j
m - j
(37)
■ s ■\2n
T1
= 'T ^
^ m=l
1.
The particular solution (35) is the critical component of the solution (34). With known, calculation of
E; and Ft is straightforward for any perturbed boundary conditions (7b,c). , ■ r •
^ Equations (27-37) provide exact, closed-form expressions for both the eigenvalue and eigenfuncuon
perturbations through second order. Their simplicity is remarkable given that they apply for an arbitrary
deviation in boundary shape. . ^ •
3rd Order Perturbation: The third order eigenvalue perturbation (26c) is
27t
K = - /( M , 5 ) = J [ ' (1/6) )lr=l ^ 0
(38)
Derivation of a closed-form expression for K is analogous to (33) and straightforward. The value of the
expression is minimal, however, because (38) can be evaluated easily using computer algebra software for
a specified g (6). Little insight can be gained from the algebraic details at third order perturbation.
The third order eigenfunction perturbation ^ ( r , 0) is
s = e u + (39)
A particular solution for the Kw term in (17a) is known by analogy with (35)
(40)
(47C)''^^CD j i(CJ))
A particular solution associated with the riv term of (17a) is
- TT £ cos;e + Cj sin/e]
Finally,
5P =
(41)
(42)
As in (30) and (36),
= £yy(cor)[//jCOs;0 + Ly siny0] (43)
M
where numerical evaluation of Hj and Lj for specified g (0) is readily achieved, e in (39) is found from
(11c)
e = -<v,w> -<u,s^> (^^4)
4th Order Perturbation: The fourth order eigenvalue perturbation % is found from (26d)
X= -T\d + (45)
1490
The fourth order eigenfunction perturbation ? ( r , 0) is
t = f u + t'' + tP
Expansion of (18a) yields
-V^t -oy^t ^{x + T]d)u +Kv^' +T\w^' +r\wP
A particular solution of (47) is
tP =ti +t^+ti+ fP^
(X + ^d) r
(Anf^w J i(t^)
-
= -
Kr
2co,tl
£ Jj^ii(£>r)[Bj cosy e + Cj siny 0]
1? = - ^ i; Jj^iiar) [ Ej oos j 6 + Fj sin; 6]
(46)
(47)
(48)
(49)
(50)
(51)
tP. = -
jfr
[cor/oCwr) - 2^1(0))]
(52)
is the only ’new’ particular solution not determined by analogy with previous particular solutions.
Jr . •J. Ji 1 1 . j /- _ /lOL'v TT _ /I
Coefficients of are calculated from (18b). The normalization (1 Id) gives
f = - V2< w,w>-<v,s>~<u,tP>
5th Order Perturbation: From (26e), the fifth order eigenvalue perturbation a is
a = - Tie -Kd + -J{u,z)
(53)
(54)
Degenerate Eigenvalue Perturbation j j i j-
Consider perturbation of a degenerate unperturbed eigenvalue C0^„ and the associated n nodal diam¬
eter orthonormal eigenfunctions (21) (the subscript mn will henceforth be omitted). Because of the
eigenvalue degeneracy, the unperturbed eigenftinction u is an element in the linear space spanned by u
and
u-a^u^+a^u^ (55)
Cc and are determined subsequently. The normalization <u,u> ~ 1 requires
+ (56)
Boundary condition asymmetry splits the degenerate unperturbed eigenvalue and fixes the coefficients <3^
and in (55). These effects might occur at first order perturbation, though, if not, they are predicted at
some higher-order perturbation. ^ j- •
Because and are solutions of the homogeneous forms of (15-19), two solvability conditions
must be satisfied at each order of perturbation. We follow the method of Parker and Mote [10].
1st Order Perturbation: The solvability conditions for (15) yield
£2^11= -/(mSv) £2^ |1= -y(M^v) (57)
Evaluation of (57) yields a symmetric, algebraic eigenvalue problem
1491
- 0)^
Sin Sin
Sin ~ Sin
= [1
Da = |xa
\il,l=^±^His^^r^(sLf}
l-iVz
(58)
The eigenvalues of D are the first order perturbations of ©. The eigenvectors of D fix the coefficients in
(55). The degenerate eigenvalue splits into distinct eigenvalues as a result of boundary asymmetry if and
only if^the fi are ^stinct. The magnitude of jl for an n nodal diameter eigenvalue is proportional to
[ is In) + iSin^i- This leads to the splitting rule: If either or both of g2n and g2n are nonzero, the n
nodal diameter eigenvalues split at first order perturbation; otherwise the eigenvalues remain degenerate.
When no first order splitting occurs, = 0 but and remain undetermined.
The eigenfunction perturbation v ( r, 6) is decomposed as
V = Cj. + Cj «■*' + V* + (59)
where the first two terms result from the two independent solutions of the homogeneous form of (15). Par¬
ticular and homogeneous solutions of (15a) are
4+l(C0r)
(27r)'/^co ^+i(CD)
( a^. cos nQ + sin n 0)
(60)
2 7y (cor) [By cos ; 9 + Cy sin j 0 ]
J=0J*n
where the j = n term of (61) is included in the first terms of (59). Using (15b),
(61)
Bj-
Cj =
CO
(27t)^Jy(CO)
(2n)%(cy)
^dSj-^n Sj~n) ^s(Sj+n ~ Sj-n)
j>n
^ciSj+n Sn~d ^siSj-i-n ■*" 8n-j)
j<n
(62a)
Sn + Sn
o
II
•'"J
^cisUn +Sj-n)-^s(sUn 'Sj-n)
j>n
(62b)
^ciSj+n ~ Sn—d ~ ^siSj-^-n ~ Sn—j)
j <n
V.
Coefficients c^. and completing the solution (59) are calculated at second order perturbation, just as
and of (55) are calculated at first order perturbation.
2nd Order Perturbation: The two solvability conditions for (16) and (11a) yield
iH^Sin
Cc
ara, 0
Tl/OO^
+ 1)
lap-
|X^(n + 1)
203^
Vijn + 1)
2
~J{u^ ,w)
- f{u\w)
(63)
1492
(0^
+ a J - CO^C ‘yiGi,) £ u- Pj j- = a, X + a, y
jj+m .
j^.i*n
•/;(C0)
j=0J^n
y,cco)
(64a)
•'wW.
,2 OO
^ j^J*n
/;(C0)
(64b)
(gUn + Sj-nf + igj+n + Sj-nY
P; j
~gj+ngj-n gj-ngj+n
i>n
aj =<
(gj+n + 8n~jY + (.gUn - gn-jY
gj+ngn-j gn-jgj+n
j<n
[2{sD-
gngn
j=o
igj+n ~ gj-n^ igj+n ~ gj~n^
gj+ngj-n gj+ngj-n
j>n
igUn - Sn-jY + (g/+„ + gLjY
J>}
il
gj+ngn-j ~ gj+ngn-j
j<n
'lig^nY
j=0
6,- = (l/4)( tty - Tj ) is used in (66). The operator in (63) is invertible if and only if the unperturbed eigen¬
value splits at first order perturbation; c^, c^, and Tj are calculable from (63). If g2n = gin ~
|J. = 0, the operator in (63) is singular, and a^. and are unknown. In this case, the component equations
of (63) yield
f{ u\w) - J{u^ ,w) = Y (a^.^ - + (Z - X)a(. <2^=0
Ti = - f{u^ ,w) - f(u\w) = -X -2Y -Z
(65)
where X, 7, and Z are defined in (64). The first of (65) and (56) can be solved for two unique
&- {Uf. a^Y if and only if one or both of the following inequalities hold
y^o-4‘/4GL+ X
j=OJ^n -gx
X ;.z ^ yaGi, + E [;• - 6y ^0
j=0J*n
Ti is then calculated from (65b). Equations (66) are second order eigenvalue splitting rules: If either or
both of (66) are satisfied, the n nodal diameter eigenvalues split at second order; otherwise they do not.
When 7 = 0 and X = Z, (65b) and (56) yield r\= - X = - Z while (65a) is identically satisfied. Thus,
r\ is calculable despite the lack of second order splitting and the continuing indeterminacy of
sequel, we assume the degenerate eigenvalues split at first order. If they do not, the development described
by Parker and Mote [11] is required.
The second order eigenfunction perturbation w ( r , 6) is
w = dc + vv'' + wP (67)
1493
wf = - - - ; — — [ (M-c^ + ria^ ) cos « e + (|a.c^ + '{\a, ) sin n 6] (68)
(27U)^^C0
= i y,.^.,(0)r)[B;Cosye + Cysin;e] (69)
w? = - - - [(HrJ-im) -2(n + l)/„+i(cor)](ac cosne + sinnO) (70)
(327c)/^coV„+i(co)
= wf +>1^1+1^^ (71)
w'' = X J;(©r)[£y cos;e + F; sinjO] (72)
j=0J^n
Components wf are associated with the three inhomogeneities of (16a) resulting from an expansion analo¬
gous to (47). The particular solution (71) is the essential element allowing calculation of the E; and F;
from (16b).
3rd Order Perturbation: The solvability conditions for (17) and (lib) give
(1 +
ay^ac
r "
4
-\i<u'^,W^> -T\<U*^ ,V> -J(u^ ,s)
<^Si,
(0^a_y
4
h ^ M
- \i<u\wP> -r\<u^ ,v> - f(u\s) (738
0
k/co^
- C0“[y2<V,V> + Oc <U^ ,W^> + <M'^, W^>]
The operator in (73) is identical to that in (63), and the assumption of first order eigenvalue splitting
ensures its invertibility in the calculation of d^yd^, and K.
The third order eigenfunction perturbation j ( r , 0) is
s +esU^ + 5^' + sP (74)
f /ri + l(d5l‘)
^ - — -—-[{\id^ +T]Cc +Ka^)cosnQ + {lLd, + Tic^ 4-K^2jsinn0] (75)
(271) ^+i(0))
X COS70 4- Cj siny'e] (76)
- [cory„(C0r) - 2(n + lV„+i(®r)](ac cosnO + a, sinnO) (77)
(327t)/W„.,i(co)
(78)
(79)
(80)
1494
_ - + ?) - [r/ (cor) + +l)_j (a)r)](a^ cosnS + a, sinnO) (81)
(288jc)’Vy„^.,((B) 2“
sP = S^l + + >5'^
(82)
s'^ = £ 7;(COr)[H,cos;0 + L;sin;0]
j=0,j*n
(83)
where the Hj , L; follow from (17b).
4th Order Perturbation: From the solvability conditions for (18) and (1 Ic)
(irgln
-
r
0
X/w^
- -T|<mSw> - k<m‘^,v> -/(m^,0
- -rl<M^W> - K<W^V> - /(M^^)
-(i?{<v,w> + ac<u\sP>-^ as<u\sP>],
(84)
The presented boundary perturbation method applies for general boundary conditions of the form
(7b, c). For annular domains, the Bessel function y„ (C0^„r) is included in (20, 21); particular solutions
associated with this additional term are almost identical to those associated with /„(co^nt- ) [10]. If dif¬
ferent unperturbed boundary conditions are considered, the unperturbed eigenfunctions (20, 21) do not
change form; only the normalization coefficients change. Consequently, the form of the right-hand side of
(15a) is unchanged, and the particular solutions (29, 60) apply £xcept for a change in their leading
coefficients. Different first order boundary condition perturbations C and C change only the values of the
coefficients Bj and Cj in (30, 61). Instead of (15b), these coefficients are determined by the general per¬
turbed boundary condition (7c)
[ v" + yl!]3P = - Ck - [ v'’ + P,, v?hp (85)
Calculation of B; and C; is always possible by Fourier expansion of (85). Thus, the forms of v in (28)
and (59) are unaffected by changes in either the perturbed or unperturbed boundary operators. As a result,
only coefficients of the particular solutions (35, 71) change for different bound^ conditions. This reason¬
ing extends to higher order perturbations, and consequently the prepnted particular solutions admit exact
eigensolution perturbations for general boundary condition perturbations.
Example Problems
The numerical accuracy achievable by the presented method is illustrated by modeling elliptical and
rectangular domains with a circle.
Elliptical Domain , , ,
An elliptical domain of eccentricity e = {I • is described by
' , V/2
1 - e-
R - a
1 - e^cos^0
(86)
where a and b are the semi-major and semi-minor axes, respectively (Fig. 2). The. average radius R (3b)
and the Fourier coefficients of g (0) (24) are calculated by quadrature. Though R depends on a and e ,
g(0) depends only on e . For an ellipse, g/ = 0 for j odd and g/ = 0 for all j . Eight non-trivial terms
through g 16 were used in the calculations.
The dimensionless, fundamental, elliptical domain eigenvalue (2) evolves from the funda¬
mental circular domain eigenvalue. (We identify the perturbed domain eigensolutions using subscripts mn
denoting the number of nodal circles m and nodal diameters n in the circular domain eigensolutions firom
which the perturbed eigensolutions evolve.) Table 1 compares the fundamental eigenvalue predicted by
1495
perturbation to the exact values computed by Daymond [13]. A maximum error of 0.32% is calculated for
eccentricities through e = 0.9 b/a - 0.4359. For the extreme eccentricity
e - 0.961 \-^b/a - 0.28, the perturbation results degrade substantially. Even without comparison with
a known solution, the degradation is evident by the poor convergence of the perturbation with increasing
order. For e = 0.961 1, llie asymptotic expansion (9) for the fundamental frequency is
©2 = 5.7831 + 0 + 5.1154 - 5.3966 + 4.7782 + 1.7258 (Qqo^) = = 7.2790
Ra=\ 0.47603
In contrast, for € = 0.9 _
= 5.7831 + 0 + 1.6774 - 0.6083 + 0.3098 - 0.0005 (floco ) = -f- = lUtu ^
The agreement between perturbation and the exact values is illustrated in Fig. 3. Results for perturbation
of the one nodal diameter circular domain eigenvalue are also given in Fig. 3, where the exact values were
obtained by optically scanning Fig. 1 of Troesch and Troesch [14] and digitizing points through e = 0.9.
Differences between the results so obtained and the perturbation values are all less than 3%, which is
approximately the precision of the scanned and digitized results. Tabular results for the (i2oi^)l,2 eigsR"
values are presented in Table 1.
Rectangular Domain
Consider the rectangular domain of dimension 2a x2b where q = b/a-< 1 (Fig. 4). The average
radius R and the Fourier coefficients of $ (0) are calculated by quadrature. R depends on a and q, but
g (6) depends only on Also, gj = 0 for j odd and g/ = 0 for ail j . Ten non-trivial terms through g20
were used in the calculations (Fig. 4).
Table 2 compares the fundamental eigenvalue perturbation to the exact value for
1 >0.3. Comparisons are shown for first through fifth order perturbation approximations. For a fifth
order perturbation approximation, errors in the fiindamental frequency are less than 0.5% for ^>0.6, 1.3%
for ^ = 0.5, and 4.7% for ^ = 0.4. For ^ = 0.3, the error is 22.% and perturbation is not effective. The
behavior of the asymptotic approximation (vertical column of Table 2) reveals a large expected error even
in the absence of a known solution.
Substantial improvement in the predicted fundamental eigenvalue results when the perturbation is
extended from first to second order. The accuracy obtainable from a second order perturbation is
significant because the closed-form expression (33) gives Tj for an arbitrary shape perturbation. Improved
accuracies are achieved with third order and fourth order perturbations. For rectangular domains, fifth
order perturbation affords no increase in accuracy. Though the accuracy achieved using fourth or fifth
order perturbation may not be needed, the higher order perturbations develop confidence in the conver¬
gence of the predicted eigensolutions through the decreased magnitude of the higher order terms.
Comparisons of perturbation predictions and exact eigenvalues are shown in Table 3 for rectangular
domains. For square domains, all predicted values differ from the exact values by less than 1%, and the
agreement is also excellent for ^ = 0.95 and ^ = 0.9, The lowest four eigenvalues provide excellent esti¬
mates for ^ = 0.7, 0.8, as shown in Fig. 5. Where the predicted and exact values differed substantially, the
failure of the higher order perturbations to approach zero in the asymptotic expansion is evident.
It is interesting that two distinct circular domain eigenvalues can merge to form a degenerate eigen¬
value pair on a square domain. For instance, the ( m , n ) = ( 1, 0) circular domain eigenvalue and one of
the degenerate ( m , n ) = (0, 2) circular domain eigenvalues merge to form the degenerate eigenvalue pair
ila =4.9673 in the square domain. In contrast, the degenerate eigenvalue pairs Q.a = 3,5124 and
Q.a = 5.6636 in the square domain evolve from the degenerate eigenvalue pairs (m,n) = (0, 1) and
( m , « ) = ( 0, 3) in the circular domain. Perturbation predicts the splitting of the degenerate square
domain eigenvalues (Fig. 5).
Conclusions
1. Eigensolutions of the wave equation with perturbations of the boundary conditions are derived by exact
solution of the sequence of perturbation problems through fifth order. Perturbations of the domain from
circular and variation of boundary condition parameters along the boundary curves are included in the class
of perturbations for which the method applies. Exactness of the perturbation solutions means no approxi¬
mation is introduced other than truncation of the asymptotic series (9, 10),
1496
2. The derived solution offers a combination of analytical and computational advantages:
• exact perturbation through fifth order yields excellent accuracy for perturbations of substantial magni¬
tude (such as the elliptic^ and rectangular domain perturbation examples);
• Fourier representation of the perturbations allows treatment of general continuous or discontinuous
asymmetries;
• algebraic simplicity of the results permits convenient use of the eigensolutions in applications such as
inverse and forced response problems;
• admissible functions as required in Ritz-Galerkin analysis are not needed;
• results are easily derived and verified using computer algebra software.
3. Rules governing splitting of the degenerate unperturbed eigenvdues are derived at both first and second
orders of perturbation. These rules take simple algebraic fonns in terms of the Fourier coefficients of a
general asymmetry. The rule for first order eigenvalue splitting is such that it can be applied by inspection.
1 J. R. Kuttler and V. G. Sigillito, “Eigenvalues of the Laplacian in Two Dimensions,” SIAM Review,
vol. 26, pp. 163-193, 1984.
2. J. Mazumdar, * ‘A Review of Approximate Methods for Determining the Vibrational Modes of Mem¬
branes,” Shock and Vibration Digest, vol. 7, pp. 75-88, 1975.
3. J. Mazumdar, ‘ ‘A Review of Approximate Methods for Determining the Vibrational Modes of Mem¬
branes,” Shock and Vibration Digest, vol. 1 1, pp. 25-29, 1979.
4. J. Mazumdar, ‘ ‘A Review of Approximate Methods for Determining the Vibrational Modes of Mem¬
branes,” Shock and Vibration Digest, vol. 14, pp. 11-17, 1982.
5. S. B. Roberts, “The Eigenvalue Problem for Two-Dimensional Regions with Irregular Boundaries,”
Journal of Applied Mechanics, vol. 34, no. 3, pp. 618-622, 1967.
6. D. D. Joseph, “Parameter and Domain Dependence of Eigenvalues of Elliptic Partial Differential
Equations,” Arch. Rat. Meek. Anal., vol. 24, pp. 325-351, 1967.
7. J. Mazumdar, “Transverse Vibration of Membranes of Arbitrary Shape by the Method of Constant-
Deflection Contours,” Journal of Sound and Vibration, vol. 27, pp. 47-57, 1973.
8. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, 1953.
9. A. H. Nayfeh, Introduction to Perturbation Techniques, pp. 426-431, J. Wiley & Sons, 1981.
10. R. G. Parker and C. D. Mote, Jr., “Exact Boundary Condition Perturbation Solutions in Eigenvalue
Problems,” Journal of Applied Mechanics, vol. 63, pp. 128-135, March 1996.
11. R. G. Parker and C. D. Mote, Jr., “Exact Higher-order Boundary Condition Perturbation in Eigen¬
value Problems,” Journal of Applied Mechanics, in preparation 1996.
12. R. G. Parker and C. D. Mote, Jr., “Exact Perturbation for the Vibration of Almost Annular or Circu¬
lar Plates,” Journal of Vibration and Acoustics, vol. 118, pp. 436-445, July 1996.
13. S. D. Daymond, ‘ ‘The Principal Frequencies of Vibrating Systems with Elliptic Boundaries,’ ’ Quart.
Joum. Mech. and Applied Math, vol. VIII, pp. 361-372, 1955.
14. B. A. Troesch and H. R. Troesch, “Eigenfirequencies of an Elliptic Membrane,” Mathematics of
Computation, vol. 27, pp. 755-765, 1973.
II
O
e = 0.5
e=0.6
e=0.7
II
o
bo
ns
It
O
VO
e =0.9611
Exact
2.5165
2.5968
2.7202
2.9215
3.2933
4.2151
6.2432
Pert. (5)
2.5165
2.5968
2.7202
2.9215
3.2936
4.2287
7.2790
% Error
0.00%
0.00%
0.00%
0.00%
0.01%
0.32%
17.%
(Doia)i
Pert. (4)
3.9212
3.9864
4.0878
4.2559
4.5726
5.3318
(f2oi^)2
Pert. (4)
4.0956
4.2822
4.5646
5.0154
5.8259
7.7601
Table 1: Comparison of the fundamental elliptical domain eigenvaluecomputed using pertur¬
bation to the exact solution of Daymond [13]. Perturbed eigenvalues evolving from the 0
nodal circle, 1 nodal diameter circular domain eigenvalues are also presented, e denotes the
eccentricity of the ellipse. Numbers in parentheses indicate the order of perturbation.
1497
^=1
^=0.9
^=0.8
^=0.7
4=0.6
4=0.5
4=0.4
4=0.3
Exact
2.2214
2.3481
2.5145
2.7391
3.0531
3.5124
4.2295
5,4665
Pert. (5)
% Error
2.5183
0.15% .
2.7460
0.25%
3.0665
0.44%
3.5597
1.3%
Pert. (4)
% Error
2.5109
-0.14%
2.7325
-0.24%
3.4797
-0.93%
4.1413
-2.1%
4.9888
-8.7%
Pert. (3)
% Error
2.2127
-0.40%
2.3394
-0.37%
3.4672
-1.3%
4.0995
-3.1%
Pert. (2)
% Error
2.2479
1.2%
2.3753
1.2%
n
3.0923
1.3%
5.9903
9.6%
Pert. (1)
% Error
2.1430
-3.5%
2.2608
-3.7%
2.4049
-4.4%
2.5859
-5.6%
2.8209
-7.6%
3.1400
-11.%
3.6013
-15.%-
4.3353
-21.%
_
Table 2: Comparison of the fundamental rectangular domain eigenvalue computed using
perturbation to exact values. ^ = b/a denotes the aspect ratio of the rectangle. The numbers
in parentheses indicate the order of perturbation.
(m, n)
(0,0)
(0, l)i
(0. 1)2
(0,2)2
(0,2)i
(1,0)
(0,3)i
(0, 3)2
Exact
2.2214
3.5124
3.5124
4.4429
4.9673
4.9673
5.6636
5.6636
^=1
Pert.
2.2243
3.5081
3.5123
4.4322
4.9679
4.9656
5.7064
5.7071
% Err.
0.13%
-0.12%
0.00%
-0.24%
0.01%
-0.03%
0.76%
0.76%
Exact
2.2806
3.5502
3.6610
4.5613
4.9941
5.2032
5.7569
5.8716
4 = 0.95
Pert.
2.2836
3.5442
3.6553
4.5499
5.2027
5.8693
5.8873
%ErT.
0.13%
-0.17%
-0.16%
-0.25%
0.00%
2.0%
0.27%
Exact
2.3481
3.5939
3.8278
4.6962
5.0252
5.4665
5.8644
6.1062
4 = 0.9
Pert.
2.3511
3.5879
3.8218
4.6831
4.9148
5.5454
6.1071
6.1390
% Err.
0.13%
-0.17%
-0.16%
-0.28%
-2.2%
1.4%
4.1%
0.54%
Exact
2.5145
3.7047
4.2295
5.0290
6.0963
6.1342
4 = 0.8
Pert.
2.5183
3.6984
4.2219
5.0063
3.9446
6.6318
6.7169
% Err.
0.15%
-0.17%
-0.17%
-0.45%
-1.9%
-35.%
8.1%
0.61%
Exact
2.7391
3.8607
4.7549
5.4783
5.2194
6.9128
6.5076
7.4289
4=0.7
Pert.
2.7460
3.8493
4.7478
5.4315
8.7571
imag.
6.1737
7.0269
% Err.
0.25%
-0.29%
-0.85%
68.%
-5.1%
-5.4%
Table 3: Comparison of rectangular domain eigenvalues from perturbation to exact values. ^ = b/a
is the aspect ratio, m and n are the numbers of nodal circles and nodal diameters in the circular
domain eigenfunction from which the corresponding rectangular domain eigenfunction evolves.
1498
Q a
nnn
Figure 3: Elliptical domain eigenvalues. The subscript mn denotes the number of nodal circles (m)
and nodal diameters (n) in the circular domain eigenvalue, a is the semi-major axis of the ellipse. The
solid lines are from the exact solutions of Daymond [13] and Troesch and Troesch [14]. The symbols
are values predicted by perturbation.
1499
Figure 4: Rectangle of aspect ratio ^ = 0.9. The
approximate rectangle is a 10 term Fourier approx¬
imation. The circle is the domain for perturbation.
Figure 5: Eigenvalues of a rectangle with aspect ratio ^ = b/a. Solid curves denote exact
values. Symbols denote values predicted by perturbation.
1500
SUBSTRUCTURING FOR SYMMETRIC SYSTEMS
A.V. Pesterev
State Institute of Physics and Technology
13/7, Prechistenka St., Moscow, 119034 Russia
E-mail: sash.a@pesterev.msk.ru
ABSTRACT
It is known that the problem of finding the spectrum of a complex
conservative system with interaction of finite rank between its subsystems
can be reduced to investigation of some symmetric characteristic matrix.
If the system exhibits symmetries of one kind or another, the problem can
be further simplified. It is shown how one can decompose the problem
for a symmetric system by using results of the representation theory of
groups. Considering a group of rigid symmetries of the system and its
representation in the interaction space, one can obtain the spectrum of the
system by investigating a number of reduced characteristic matrices, the
sum of their orders being equal to the rank of the interaction.
1 INTRODUCTION
It is well known that the problem of finding dynamic characteristics
of a complex structure consisting of distributed and/or finite- dimensional
subsystems which interact at a finite number of points can be reduced to
investigation of some matrix of relatively small order (see, e.g., [1-10] and
references therein), which is sometimes referred to as the characteristic
matrix. If the system exhibits symmetries of one kind or another, the
problem can be simplified from the computational standpoint. The aim
of this paper is to develop a technique for investigation of such systems.
To facilitate discussions, we shall restrict our consideration to conservative
systems. To reduce the original spectral problem to the problem of inves¬
tigation of a characteristic matrix, we will apply the technique used in the
structural analysis method [5, 6].
1501
2 FORMULATION OF THE PROBLEM
Let us consider a linear complex mechanical system consisting of a finite
number of distributed and/or finite-dimensional subsystems interacting at
a finite number of points. Let us write the equation governing small sta¬
tionary vibrations of jth isolated subsystem in the following operator form
Aj{X)Xj = Fj, (1)
where Fj is a vector function of amplitudes of external harmonic general¬
ized forces acting on the subsystem, Xj is an amplitude response vector
function, A = w is a circular frequency, Aj(A) is a self-adjoint operator.
In the case of a distributed subsystem, Aj(A) is a differential operator; for
a finite- dimensional subsystem, it is given by a matrix.
Let the subsystems interact elastically at a finite number- of points
(e.g., they are connected to each other by means of a finite number of
conservative springs). Let us denote by X the i?-component vector func¬
tion of responses of the system, the jth component of X being vector
function Xj. Clearly, the interaction forces depend on displacements of
only those points that take part in the interaction; i.e., they depend on
some finite-dimensional vector Y e rather than on response vector
function X, which, in the general case, belongs to some functional space.
Denote by S an operator transforming X into the vector Y: Y = SX. Let
the interaction be given by a stiffness matrix K such that the amplitude
vector of generalized interaction forces F C R^ is given by the equation:
F = —KY. As the interaction is assumed to be conservative, the matrix K
is positive semidefinite. To transform vector F into the right-hand sides
of equations (1), the adjoint operator S* is used (see [5, 6] for more de¬
tail). Assuming zero external forces, one arrives at the following equation
governing free stationary vibration of the system under consideration
(A(A) + S*KS)X = 0 , (2)
where A(A) is a diagonal operator matrix A(A) = diag[Aj(A)<5jjfe]j^j;._i gov¬
erning vibration of the aggregate of the non-interacting subsystems. The
discrete spectrum of the system is defined to be the set of numbers A such
that equation (2) has non-trivial solutions. More detailed discussions con¬
cerning such a formulation of spectral problems, as well as some examples,
are given in [5, 6].
It is well known (e.g., [1, 3-6]) that if operator A“^(A) can be calcu¬
lated, the problem of finding the spectrum can be reduced to the problem
of investigating some characteristic matrix. The matrices can be defined
differently, but all of them have an important property: if A~^(A) exists,
problem (2) has non-trivial solutions if, and only if, the characteristic ma¬
trix is singular.
1502
In this paper, we will take advantage of the definition of the charac¬
teristic matrix used in the structural analysis method [5-7]. As the stiff¬
ness matrix K is positive semidefinite, it ^n be factored in the form [11]:
K = a, where a is Ny.N matrix, and N is the rank of matrix K. The
characteristic matrix of order N is then defined by the equation
Q(X)^ I + aSA-\\)S*a^ . (3)
Properties of eigenvalues and eigenvectors of the matrix that guarantee
finding all eigenfrequencies of the system in a given frequency range are
discussed in [6]. To save room, we will not discuss here the ways of in¬
vestigation of those values of A at which A"^(A) does not exist (detailed
discussions of this case as applied to nonsymmetric structures can be found
in [8, 9]), as this has no effect on the decomposition discussed.
Let now the subsystems (or part of them) be identical or can be divided
into a few groups of identical ones and interact in such a way that the com¬
plex system exhibits symmetries of one kind or another. It is reasonable
to suggest that these symmetries can be used to simplify the problem and
also understand and classify solutions. The technique to be discussed in
this paper allows one to decompose matrix (3) into a number of matrices
of lesser orders such that the spectrum of the system can be obtained by
investigating these matrices separately.
3 DECOMPOSITION OF THE INTERACTION
SPACE
Let Cj be a group of symmetry of the immovable system, elements of
the group g ^ G being either rotations or reflections that transform the
system into itself. Clearly, as we consider systems with a finite number of
interaction points, G can be only a finite group. For the same reason, we
can consider a representation of the group in a finite-dimensional space.
Actually, it follows from the results discussed in [5, 6] that at a given eigen-
frequency c<;o J^ere is one-to-one relationship between eigenfunctions of the
system and A/’-dimensioi^l vectors from the kernel of the matrix Q{Gl)
KerQ(a;Q). However, as A/’-dimensional space, come into existence as a re¬
sult of a formal procedure of the factorization of the stiffness matrix A, it
may be difficult to build a representation of the group in this space. There
are two physical A-dimensional spaces relevant to the problem, namely,
the space of displacements of the points at which the subsystems interact
with each other and the space of generalized interaction forces. Either of
them can be used to build a representation of the group. Let, for definite¬
ness, it be the space of displacements Y = SX\ we denote it by letter L
and will call the “interaction” space. It is evident from general consider¬
ations that at a given eigenfrequency there exist one-to-one relationship
1503
between the eigenfunctions of the system and vectors of the corresponding
displacements of the connection points.
Let the system free vibrate at some eigenfrequency, and let y 6 L be
the corresponding vector of the displacements of the connection points.
From this point on, we will call it, for brevity, the “eigenvector” of the sys¬
tem. Any rotation or reflection g e G transforms the eigenvibration into
some, generally speaking, different eigenvibration, the eigenvector Y being
transformed into some other eigenvector Let us introduce linear oper¬
ators (matrices) T{g) by the equation: T{g)Y = g Clearly, T(g)
is a real unitary representation of group G and can be constructed for any
particular configuration of the interaction points and a given coordinate
system.
As will be seen from the next section, the decomposition of the spectral
problem can be obtained if we succeed in decomposing the space L into a
number of mutually orthogonal real subspaces
L = Li -f • ■ • + Ljn (4)
such that any eigenvector of the system Y could be represented in the form
y — where Yj e Lj is also an eigenvector of the system corresponding
to the same eigenfrequency. In other words, we need such a decomposition
that either an eigenvector Y belongs to a subspace Lj or (in the case of
a multiple eigenfrequency) there exist such elements of the group gk and
real numbers (Sk^ which are not all zero, that Ylkl^kT{gk)Y G Lj. Because
of lack of room, we restrict our discussions to substantiating reasoning and
give the results without proofs.
Let Ti{g), . . . , Tq[g) be real nonequivalent irreducible unitary represen¬
tations of the group G and representation Tj{g) of dimension Vj occurs rrij
times in T{g). The following lemma is valid.
Lemma 1 For any gwen j , there exist such g^ G G and real numbers 13k,
/c = 1, . . . , Tj + 1, that Ylk KTj{gk) = 0.
The proof is based upon application of Cayley-Hamilton theorem [11] to
the matrix Tj{g) with g being an element of a cyclic subgroup. It is well
known [12] that the space L can be decomposed into mutually
orthogonal invariant subspaces each of which is transformed by one of
the irreducible representations. Let us denote by Lj {j = 1,...,^) the
direct sum of all invariant subspaces transformed by the representations
equivalent to the real representation Tj{g). Using Lemma 1, one can prove
the following theorem.
Theorem 1 For any eigenfrequency, an eigenvector Y either belongs to
one of the suhspaces Lj or can be represented in the form Y = YlYj, where
Yj G Lj is also an eigenvector of the system. If an eigenvector of the
system belongs to a suhspace Lj, then the multiplicity of the eigenfrequency
is equal to rj.
1504
Thus, we have obtained the desired decomposition (4) of space L with m
being equal to the number of real nonequivalent irreducible representations
of point group G. To take advantage of decomposition (4), one needs
projectors Pj onto the subspaces Lj : PjL = Lj. The projector on the
subspace Lj is given by the formula [12, Chapter 4]
= (5)
g^G
where n is the order of the group G, and Xi(5') is the character of Tj{g).
It turns out that under certain conditions some of subspaces Lj, which
are transformed by representations whose dimensions are greater than one,
can be decomposed in turn.
Theorem 2 Let Tj{g) be a real irreducible representation of a group G of
dimension rj > I, and let there exist an element of the group go of order
T-j such that the matrix Tj{go) has all different complex eigenvalues. Then
the subspace Lj can he decomposed into Pj subspaces, where, for even r^,
Pj = rff2 + 1, and, for odd rj, pj = {rj + l)/2.
The theorem allows one to obtain more subtle decomposition (4) with m
being greater than the number of real irreducible representations.
Let us obtain formulas for projectors onto the subspaces of a decompos¬
able subspace Lj. As mentioned above, Lj is the direct sum of subspaces
L), k = l,...,mj, transformed by the representations equivalent to Tj{g).
Let us consider for a while Lj as complex subspaces. It is evident that
eigenvalues of the matrix Tj{go) are numbers (*, i = 0, 1, . . . ,rj - 1, where
C is a primitive rf root of 1 = 1). Let e\ be the corresponding eigen¬
vectors of matrix Tj{go) belonging to the subspace L^. If rj > 2, some
vectors ef are complex, and, for each complex vector ef , there is its com¬
plex conjugate vector ef. Denote by ^j • (z = 0, 1, . . . , Pj — i) the subspace
of Lj spanned either by vector ef, if is real, or by two vectors ef and
ef. Denote by Ljj the direct sum of subspaces /j^^, k = l,...,mj. It is
the decomposition of Lj into pj real subspaces Lj^i that is established in
Theorem 2.
Let us take the above eigenvectors e-’’, i = 0,l,...,rj — 1 for the or¬
thonormal basis in subspace Lj and write the representation Tj{g) in ma¬
trix form Tffg) = z, / = 0, 1, . . . , rj - 1, which is the same in all
subspaces Lj, fc = 1, . . . ,mj. Using the results of [12, Chapter 4], one can
obtain formulas for projectors Pj^i on the subspaces Lj^i
Pj- = i'MTig) , Z = 0 and z = p, - 1 (for even rj) , (6)
g^G
Pj.i = - Z(«y<7) + iU9))T{9) .otherwise. (7)
^ geG
1505
Note that the eigenvectors of the system Yj,i 6 Lj^i exist or do not exist
in all subspaces Lj^i simultaneously, so that the multiplicity of the corre¬
sponding eigenfrequency of the system is equal to rj as before.
4 DECOMPOSITION OF SPECTRAL PROBLEM
In the previous section, we have decomposed space L of displacements
of the connection points. One could take for L the space of generalized
interaction forces and obtain the same decomposition. It is not difficult to
prove that if the system free vibrates and the corresponding eigenvector
Y belongs to a subspace Lj then the corresponding vector of interaction
forces F = —KY also belongs to the same subspace Lj. This allows one
to introduce matrices Kj = PjKPj., ^’ = 1, . . . , m, and consider m spectral
problems of the form (2) in which the matrix K is replaced by Kj] here
projector Pj is given either by formula (5) or by (6)-(7). Matrices Kj
can also be presented in the factorized form: Kj = aja, where aj is
NjxN matrix, and Nj is the rank of Kj. Let us define, analogously to (3),
matrices Qj{X) of orders Nj by the formula
Qj(X) = ! + aiSA-^X)S‘aJ . (8)
It turns out that to obtain the spectrum of the system, one can investigate
separately matrices Qj(A) instead of one matrix Q{X). Let Uj be the set
of values of A such that matrix (5i(A) exists and is singular. Denote by U
the union of the sets f/j, ; = 1, . . . , m. If some A occurs r times in C/, we
say that it has multiplicity r. __
Theorem 3 1. The sum of orders Nj ofQj{X) is equal to the order N of
QW- '
2. The set of the eigenvalues of the system different from those of the
isolated subsystems coincide with the set U .
The eigenvalues of the isolated subsystems are investigated separately. The
approach used in [5, 6, 8, 9] for nonsymmetric systems can be modified and
applied to symmetric ones.
Remark, In the general Cctse of a 3D structure consisting of several
substructures, the subsystem points that take part in the interaction can
be divided into a few independent groups, called “interaction sections,”
such that the interaction forces acting on the subsystems at the points
belonging to a certain interaction section depend on displacements of only
those points that belong to this interaction section. This subdivision of
the interaction points into independent interaction sections implies a pri¬
ori decomposition (not connected with the symmetry of the system) of the
interaction space and results in a block-diagonal form of the stiffness ma¬
trix K (under an appropriate numbering of the interaction points). This
1506
implies also that we can build representations of the group in different
interaction sections separately.
5 PROJECTORS FOR THE GROUPS AND
As an illustration, let us decompose the interaction space and calcu¬
late projectors for two cases of symmetry. First, consider a system having
n-fold axis of symmetry such that rotations about the axis through an¬
gles A: = 0, 1, . . . ,n - 1, transform the system into itself. The
rotations form the cyclic abelian group On of order n. It has [12] n
one-dimensional complex irreducible representations. Combining pairs of
complex conjugate representations into real two-dimensional representa¬
tions, one obtains m real irreducible representations, where, for even n,
m — n/2 -f 1, two representations being one- dimensional and the others
being two-dimensional; for odd n, m = (n l)/2, with one representation
being one-dimensional. Calculating the characters of the representations
and applying formula (5), one obtains formulas for the projectors onto the
subspaces Lq and Lnf2 (for even n) corresponding to the one- dimensional
real representations
P, = iy:T(Ci') . Pj = iE(-l)‘r{0 (even n) , (9)
and onto the subspaces Lj corresponding to the two-dimensional real rep¬
resentations
2 ^27rkj
Pj = - )r(cj
k=0 ^
, J =
1. . . . , I — 1, for even n,
1. . . . , for odd n.
(10)
As a second example, let us consider the group of symmetry Cnv, which
also describes symmetry arrangements with one n-fold axis. But apart
from the rotations, there are n reflections transforming the system into
itself. The order of the group is equal to 2n. Let us choose the elements
of the group as follows: Cj = e, . . . , ayCl^ . . . where
CTy is a reflection in a certain symmetry plane passing through the n-fold
symmetry axis. It is known [12] that for odd n this group has two one¬
dimensional (To,i and To, 2) and (n - l)/2 two-dimensional real nonequiv¬
alent irreducible representations. For even n, it has four one-dimensional
(To,i, To,2, Ti,!, and Tb,2) and (n - 2)/2 two-dimensional representations.
Calculating the characters of the one-dimensional representations and ap¬
plying formula (5), one obtains the projectors onto the subspaces corre¬
sponding to the one-dimensional representations
YE(T(C‘) + T(a.O)
k=0
(11)
1507
(12)
^o,2 = ^i:(r(c*)-r(c7„0),
k=0
and (if n is even)
= ;r £(-1)'= (net) + T(<r„C'i)) , (13)
fc=0
Ph^ = r £(-1)'' {net) - T(<r„C„‘)) . (14)
k=0
It is not difficult to show that bases in the spaces Lj corresponding
to the two-dimensional representations Tj{g) can be chosen in such a way
that the matrices Tj{cry) are diagonal: Tj{cry) = diag[l,— lj. This means
that the two-dimensional representations Tj{g) satisfy the conditions of
Theorem 2 with go = <7^, and, hence, each of the subspaces Lj can be
decomposed into the sum of two real subspaces Lj = Lj^ + Lj^i . Applying
formula (6), one obtains the projectors onto the subspaces Lj^o and Lj^i :
P,.0 = i E cosl"^) {T{Ct) + n<^.Ct)) , (15)
^ A:=0 ^ ^
where
Pi,i = - E cos(?^) (T(C,t) - na„Ct)) .
^ k=0 ^
J
1, . . . , for even n,
1, . . . , for odd n.
(16)
6 AN EXAMPLE OF CONSTRUCTION OF A
REPRESENTATION
In the previous section, we obtained formulas (9)-(16) for the projec¬
tors for the symmetry groups Cn and Cnv To take advantage of these
formulas for a given symmetric structure, it is required only to construct
a representation of the group in the corresponding interaction space. The
aim of this section is to demonstrate, by way of a simple example, how
such a representation can be constructed.
Let a symmetric structure consist of seven interacting subsystems, and
let its symmetry be described by the group Cey. Let the 6-fold axis of
symmetry be perpendicular to the plane of the sheet. Keeping in mind
Remark in Section 4, let us consider one interaction section of the sys¬
tem. Let the interaction points belonging to this section be located at
the vertices of a regular hexagon as shown in Fig. 1. Here the circles
denote the interaction (connection) points of the subsystems belonging to
1508
yi
1
Z2
6
o
o
2
i
7
O
5
4
O
Fig. 1: An interaction section for a system with C^v symmetry group
the considered interaction section. The kind of the subsystems (finite- or
infinite-dimensional) is of no concern at this stage, since a representation is
constructed in a hnite-dimensional interaction space. Note, however, that
the configuration shown in Fig. 1 could represent, e.g., a section of a bundle
of seven identical 3D beams by a plane perpendicular to the longitudinal
axes of the beams.
Let the points be connected to each other through some springs, and
the interaction be described by a stiffness matrix of interaction K. To
construct a representation, a particular configuration of springs is of no
importance. It is required only that this configuration satisfy the same
symmetry conditions. That is, the system of springs transforms into itself
under any rotations and reflections belonging to the considered group Cqv
To simplify the consideration and to save room, we will assume first
that the interaction points are allowed to vibrate only in the plane of the
figure, and the interaction forces also belong to this plane. Thus, the
interaction space L in this example is 14-dimensional. The construction
of the representation is simplified if we represent the displacement of each
interaction point in its own coordinate system, the origin of which is placed
at the equilibrium state of the point. The coordinate systems associated
with point 1 and central point 7 are shown in the figure. The coordinate
system of point (A: = 2, . . . , 6) is obtained from that of point 1 by rotation
about the symmetry axis x (passing through the central point) through the
1509
angle Cq = The vector of displacements of the interaction points is
given hyY = [j/i, , y?, zj]'^ e where y* and Zk are the coordinates
of the displacement of the ^th point in its own coordinate system.
As discussed in the previous section, the group Cev consists of 12 el¬
ements. Let cr„ be the reflection in the symmetry plane passing through
the 6-fold axis x and points 1 and 4, and Cq be the counterclockwise ro¬
tation about the 6-fold symmetry axis x through the angle Cq. Let the
elements of the group be chosen in the same way a.s in Section 5: Cq~
. . . .cTyCi. Since T(yiy2) = 'R{gi)T{g2)ygi,g2 ^ G
[12], we need to construct only the matrices r(C'|), = 1, . . . , 5, and T{ay).
It can be checked directly by examining Fig. 1 that the representation
T is given by
■ E 0 0 0 0 0 0 ■
0 0 0 0 0 i; 0
0 0 0 0 0 0
T{cry) = 0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 / 0 0 0 0 0
0 0 / 0 ‘0 0 0
0 0 0 / 0 0 0
T{Cl) = 0000/00
0 0 0 0 0 / 0
/ 0 0 0 0 0 0
0 0 0 0 0 0 Fi
'oo/ooool fooooo/o'
000/000 1 0 0 0 0 0 0
0000/00 0/00000
TiCl)= 0 0 0 0 0 / 0 ,..-,T(C'|)= 0 0 / 0 0 0 0 ,
1 0 0 0 0 0 0 000/000
0/00000 0000/00
000000 r2j [oooooor5_
where 0 stands for the 2x2 matrix of zeros, I is the 2x2 identity matrix,
E is the diagonal matrix E = diag[l, —1], and
Clearly, T(C'g) is the identity matrix of order 14, and the other five
matrices are obtained by means of the formula — T{cry)T{CQ).
Now, whatever is the stiffness matrix corresponding to this interaction
section, the interaction space is decomposed into eight subspaces, and the
projectors onto these subspaces are easily constructed by means of formulas
(11)-(16) at n = 6. Repeating this procedure for the other interaction
sections and applying the technique discussed in Section 4, we obtain eight
characteristic matrices such that the spectrum of the original problem can
be obtained by investigating these matrices. The orders of these matrices,
in the general case, are different and determined by a particular stiffness
1510
matrix of the interaction. However, according to Theorem 3, the sum of
their orders is equal to the order of the characteristic matrix of the system
when it is investigated without regard for its symmetry.
Note finally, without going into detail, that the consideration of a more
general case, the case of space displacements of the interaction points,
presents no additional problem. Let, for example, the vibration of a bundle
of seven 3D beams be analyzed. Let the beams be connected to each
other through some number of translational and torsional springs located
in a plane perpendicular to the longitudinal axes of the beams. Let the
interaction section of the bundle be as shown in Fig. 1. In this case, the
displacement of each interaction point (which is actually a cross-section
of the corresponding beam) is described by six (three linear and three
angular) coordinates. Introducing the third axis for each coordinate system
and the Eulerian angles, one can construct the representation in the 42-
dimensional interaction space. Writing the matrices of the representation
in terms of 6x6 matrix blocks, one will obtain them exactly in the same
form as in the previous case of plane vibrations. Blocks 0 and /, in this case,
are 6x6 zero and identity matrices, respectively; E is a diagonal matrix
of ones and minus ones; and Tk describes the coordinate transformation
under rotation about the 6-fold symmetry axis through the angle Cq. A
concrete form of matrices E and Tk depend on the chosen enumeration of
the six coordinates describing the position of the point in the interaction
space.
7 NUMERICAL ILLUSTRATIVE EXAMPLE
The aim of this section is to give a simple numerical example of decom¬
position of the interaction space. Since the result of the decomposition of
the interaction space (i.e., the number of nonempty subspaces and their
dimensions and, hence, the number of the reduced characteristic matri¬
ces and their orders) does not depend on the kind and complexity of the
subsystems and is determined only by the symmetry group, the configu¬
ration of the interaction points, and the stiffness matrix of the interaction
between the subsystems, we will consider a structure consisting of the sim¬
plest subsystems — concentrated masses.
We will consider a system possessing the symmetry group C&v and take
advantage of the representation constructed in the previous section. Let
the free-free spring-mass system shown in Fig. 2 perform plane vibration.
Here the circles denote identical masses m = 1, and the lines connecting
the masses denote linear springs of the stiffness k = 1.
The direct analysis results in a conventional matrix formulation of the
spectral problem. The eigenvalues (eigenfrequencies squared) are obtained
1511
1
2
3
4
Fig. 2: A spring-mcLSS system of 7 masses and 12 springs
by means of solving the eigenvalue problem
{K-XM)h = 0, (17)
where M is the identity mass matrix of order 14, and the stiffness matrix
K of the same order is easily constructed by means of Fig. 2. The direct
computations yield 3 rigid body and 11 elastic modes. The eigenvalues
corresponding to elastic modes are 0.849 (2), 1.000 (1), 1.229 (2), 2.000 (1),
2.651 (2), 3.000 (1), and 4.271 (2). The numbers in the parentheses indicate
the multiplicities of the eigenvalues.
The interaction space in this degenerate example is also 14-dimensional.
Denote Kj = FjKPj, where Pj is one of the eight projectors defined by
equations (11)-(16). As follows from the discussions of Section 4, spectral
problem (17) can be decomposed into the eight problems
{K,-XM)h = 0.
Factoring the matrices Kj^ Kj = ajaj, and applying equation (8), we
arrive at the eight characteristic matrices Qj(X). For this example, Qj(X) =
I - (1/A)(7j, where qj = ajaj, and the nonzero eigenvalues are found as
solutions to the eight characteristic problems: {qj — X!)z = 0.
The table below presents the dimensions of the characteristic matrices
(the third row) corresponding to each of the projectors defined by equa¬
tions (11)-(16) and the eigenvalues obtained by solving the corresponding
characteristic problems.
1512
J
1
2
3
4
5
6
7
8
Projector
■Fb,i
Po,2
^3.1
Psa
Pho
Pi,i
^2,0
■^2,1
N,
1
0
1
1
2
2
2
2
X
2.000
1.000
3.000
1.229
4.271
1.229
4.271
0.849
2.651
0.849
2.651
As can be seen, all the repeated eigenvalues of the original problem are
split now; they appear cls solutions of different eigenvalue problems (one
eigenvalue corresponds to a symmetric vibration with respect to any ver¬
tical symmetry plane passing through the 6-fold symmetry axis, and the
other corresponds to the antisymmetric vibration). Note also that the di¬
mension of the characteristic matrix corresponding to the projector Po,2 is
equal to zero. This implies that the system has no eigenvibration such that
the corresponding vector of interaction forces belongs to the space Lo,2-
In conclusion, note that though the decomposition of the spectral prob¬
lem for this simple example results in no advantages (the time required to
factor all the matrices Kj is greater than the time required to solve the
original problem (17)), the situation becomes dramatically different in the
case where complex subsystems are involved. To make the point clear,
let us imagine that Fig. 2 represents the interaction section for a system
consisting of seven complex subsystems. Let circles denote the connec¬
tion points of the subsystems. Assume that the Green’s operators of the
subsystems are available (e.g., they can be calculated by means of the
modal series). If one applies some substructuring method for investigation
of such a system, one arrives at the investigation of a certain relatively
small matrix (if the structural analysis method [5, 6] is used, the order
of the characteristic matrix will be equal to 14), which, however, depends
nonlinearly on the spectral parameter A. The finding of the system spec¬
trum, in this case, will require many iterations involving calculation of the
matrix and search for its singularity points (zero determinant). The use
of the system symmetry in this case will allow one to reduce the spectral
problem to investigation of seven characteristic matrices of orders one or
two (see the above table). Since the skeleton factorizations of the matrices
Kj are implemented only once (before the iterations), the effect of using
the decomposition is evident.
8 CONCLUDING REMARKS
The method of investigation of complex symmetric structures has been
presented. The technique is based upon results of the group representa¬
tion theory. To obtain the decomposition of the spectral problem, the
interaction space of the system is decomposed by means of real irreducible
representations of the group describing the symmetry of the system. The
1513
projectors onto the subspaces of the interaction space are given for the two
symmetry groups.
If the structure under investigation possesses the symmetry described
by the group (7„ or Cnv, in order to decompose the spectral problem, one
can take advantage of the results presented in the paper. In this case, one
needs only to build a representation of the group in the relevant interaction
space. The way of construction of the representation of the group C^y for a
particular configuration of the interaction points is presented in Section 6;
it can be easily extended to other configurations that fall in the group Cn
or Cnv
As to other kinds of symmetry, a general scheme of application of the
method discussed in the paper can be briefly summarized as follows:
(i) First, the symmetry group relevant to the investigated structure is de¬
termined.
(ii) Second, the connection points of the subsystems are separated mental¬
ly from the structure, divided into independent interaction sections, and a
representation of the group is built for all interaction sections.
(iii) Then, one should abstract from the structure and to find the decom¬
position of the interaction space. This is achieved through finding real
irreducible representations of the group. For many groups, these represen¬
tations are available in the literature. Note, however, that they are often
complex, and one should combine pairs of complex conjugate irreducible
representations to obtain real representations. The case where certain
subspaces transformed by representations of order greater than one can
be split is more difficult for analysis. To determine whether or not this is
possible, one can take advantage of Theorem 2 or try to find other criteria,
which may occur more convenient in particular cases. This stage is com¬
pleted with the construction of the projectors onto the subspaces of the
interaction space. The projectors are built by means of formal application
of formulas (5) or (6)-(7).
(iv) Finally, the reduced characteristic matrices are constructed by means
of the technique described in Section 4.
At last, note that, in the authors’ opinion, the approach discussed in the
paper can be used to develop a method for analysis of another interesting
class of structures — the structures that are not symmetric but consist of
identical subsystems (or several sets of identical subsystems). The group
relevant to such a class of systems is a subgroup of the permutation group.
We suppose that the close examination of this case from the standpoint of
the group representation theory may lead to the development of an efficient
method for analysis of this important for applications class of structures.
1514
References
1. Bergman, L.A. and McFarland, D.M, On the vibration of a point
supported linear distributed structure. J. Vibr., Acoustics, Stress and
Reliability in Design, 1988, 110, 485-492.
2. Garvey, S.D., Friswell, M.I. and Penny, J.E.T., Efl&cient evaluation
of composite structure modes. Proc. of the 5th Int. Conf. on Recent
Advances in Structural Dynamics, Southampton, July 1994, 869-878.
3. Gould, S.H., Variational Methods for Eigenvalue Problems, Oxford
University Press, London, 1966.
4. Kron, G., Diakoptics, MacDonald, London, 1963.
5. Pesterev, A.V. and Tavrizov, G.A., Vibrations of beams with oscilla¬
tors I: structural analysis method for solving the spectral problem. J .
Sound Vibr., 1994, 170, 521-536.
6. Pesterev, A.V. and Tavrizov, G.A., Structural analysis method for dy¬
namic analysis of conservative structures. Int. J. Anal. Exper. Modal
Analysis, 1994, 9, 302-316.
7. Pesterev, A.V and Tavrizov, G.A., Quick inversion of some meromor-
phic characteristic matrices. Proc. of the 5th Int. Conf. on Recent
Advances in Structural Dynamics, Southampton, July 1994, 890-899.
8. Pesterev, A.V. and Bergman, L.A., On vibration of a system with an
eigenfrequency identical to that of one of its subsystems, ASME J.
Vibr. Acoustics, 1995, 117, 482-487.
9. Pesterev, A.V. and Bergman, L.A., On vibration of a system with
an eigenfrequency identical to that of one of its subsystems, Part II,
ASME J. Vibr. Acoustics, 1996, 118, 414-416.
10. Yee, E.K.L. and Tsuei, Y.G., Direct component modal synthesis tech¬
nique for system dynamic analysis. AlAA J., 1989, 27, 1083-1088.
11. Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge Univer¬
sity Press, Cambridge, 1985.
12. Lubarskii, G.Y., Theory of Groups and its Applications to Physics,
Moscow, 1958, (in Russian).
1515
1516
RANDOM VIBRATION I
ANALYTICAL APPROACH FOR
ELASTICALLY SUPPORTED CANTILEVER BEAM
SUBJECTED TO MODULATED FILTERED WHITE NOISE
Gongkang Fu and Juan Peng
Department of Civil and Environmental Engineering
Wayne State University, Detroit, MI 48202, USA
TEL:(313)577-3842; FAX:(3 13)577-3881
SUMMARY
For elastically supported cantilever beams, this paper presents an
explicit solution of response statistics to seismic input at the base -of the
beam. The excitation is modeled by modulated filtered white noise. The
modulation is described by the sum of exponentially decaying functions,
accounting for nonstationarity of strong ground motion. The solution was
obtained by analysis in the time domain using the state space approach and
impulse response. An example is also included here to demonstrate the
application to bridge piers, in developing probabilistic seismic response
spectra for design and retrofit of bridge structures.
1. INTRODUCTION
Although various methods have been suggested for random vibration
problems, only a few explicit solutions have been reported for random
vibration under nonstationary excitations Explicit solutions are
often desirable for their advanced abilities of verifying approximation
methods and providing insights of the problem. This paper contributes to the
knowledge in this area by presenting an explicit solution for the
displacement, velocity, and acceleration of a structural system with
continuous parameters. It is subjected to strong ground motion modeled by
exponentially modulated filtered white noise. The double filter in casecade
suggested by Clough and Penzien is included in the excitation. This
solution was obtained by analysis in the time domain using the state space
approach and the impulse responses.
2. EQUATION OF MOTION
Consider a vertical cantilever beam with constant cross section. Its
horizontal displacement V(x,t) at time t and distance x from its base is
giverned by the following motion equation, with an assumption of viscous
damping:
1517
m, V”(x,t) +c, V’(x,t) + El = 0
(1)
where the number of primes to V(x,t) indicate the orders of partial
derivatives with respect to time t, and the Roman superscript denotes that
with respect to location x along the length of the beam, and c^. are
constants for mass and damping per unit length, and El is a constant
denoting the flexural stiffness of the beam.
After the displacement V(x,t) is decomposed into pseudo-static and
dynamic displacements:
V(x,t)=V3(x,t) + V,(x,t) ; V, (x,t)=f(t)(l-x/L) (2)
Eq.(l) becomes
m, V/’(x,t) + c, V,’(x,t) + El V;"(x,t) = - m, r(t)(l-x/L) (3)
It is assumed here that the damping term f (t)(l-x/L) is negligible. The
associated initial and boundary conditions are identified as follows:
v,(x,0) = 0 v;(x,0) == 0 (4)
k, V,(0,t) 4- El V,>"{0,t) = 0
k. v;(0,t) - El V/(0,t) = 0
V/‘(L,t) = 0
V/(L,t) = 0 (5)
The boundary condition at the base (x=0) in Eq.(5) describes dynamic
equilibrium involving deformations at the base. k( and k^ denote the elastic
stiffness for translation and rocking, respectively. This inclusion is intended
to model effects of the foundation. For example, deep pile foundations may
be modeled by setting k^ and k^ equal to infinity.
The ground acceleration is modeled by a nonstationary random
process, described by the product of a deterministic modulation Tj(t) and a
stationary process g(t):
r(t) = ri(t)g(t) (6)
11 (t) is described by the sum of a series of exponential functions in a general
form:
Ti(t)= Zi=,^,aiexp(bit) (t>0; ri(0)=0; ri(oo)=0) (7)
1518
where a| and b; are real constants, and bj must be negative. This general
form of modulation is used here for including several popular models and
ease of mathematical derivation. g(t) is assumed to be a white noise process
with zero mean and filtered by the Clough-Penzien filter modeling soil
effects in cascade. Its spectral density function Sgg(co) and autocorrelation
function Rgg(x) are respectively given by
0,/ + (2i;g,0g:0)^ S
Sgg(0) - - (8)
+ (2(;g,0g,0)" (cOgj'-©^)^ + (2Q2COg2to)' 271
S 1
Reg(x)= - Ik=i,2 - exp«g,0g, I x|)
4 Cgic<^gk
^gk^gk
[(^ak gdk I 1 (Cak"Ct,(.)Sm0g(jV; j X | ] (9)
“gdk
where C,;, and C^k (k=l,2) are respectively as follows;
C,, = - — [(l+8<;„^-16Q/)(l- — ) - 8^,,^ - (1-21;^^ - + 2?,,^ - )]
D COj,"* co/ co/
20gi‘ “s' “s' “'6'
Cg, = — [(i+8;;gM6i;g,‘)(i-2^g,- — + 2;;^,^— )-2q,^(i - )]
D aJ “sz'
kUg2 Wg! v^gl ^gl ^gl
Ca2 = - [(l + 8Cgi^-16Cg,'^)(l — ) - 8^g,^— (l-2i;g2^- — + 2i;g,^ - )]
D
C0g2^ C0g2'
©g2' C0g2'
®gl'
®gi"
— (-
— -8Cg,^+l6gV)(l-2^;s.-
— + 2g^ — )
(»g2'
^g2^
C0e2' C0g2-
— )]
(10a)
®g2'
and
1519
(10b)
®Sl
D = -40),,^(0,,^(l-2^„^ - + - +
0)/ CB,j" (0,j" O5/
+ (0/(l - )^
In Eqs.(8) to (10), (^g,, ^gj, cOg,, and C0g2 are characteristic damping ratios and
frequencies of the respective subfilters indicated by subscript k=l,2. cOg^k
= cDgk(l-(!^gk^)'^ (k= 1,2) are their characteristic damped frequencies. 8/271 is
a constant indicating the intensity of white noise.
Note that RggCr) is the Fourier inverse transform of Sgg(cD) :
^£8^^) exp(/coT) dco (11)
where /=(-l)‘'^ is the imaginary unit and Rgg(T) was obtained by
decomposing Sgg(co) into the sum of two terms:
S Cak COgk^ + CbkCD^
Sgg(o))= — Ik=t.2 - (12)
27: (o)gi^-coy + (2(;g,0)g,CD)"
and performing the inverse Fourier transform individually for each term by
the method of residues Note that the Kanai-Tajimi filter can be
represented in Eqs.(8) to (10) by setting C0g2=0, leading to Ca,=C0gj^,
and Ca2=Ci,2=0 On the other hand, C0gi->oo eliminates the
effect of the Kanai-Tajimi filter. Note that this seismic excitation model of
modulated cascade-filtered white noise covers nonstationarity of intensity,
but not that of frequency contents. Either or both of them may be observed
in strong motion records.
3. SOLUTION BY STATE SPACE FORMULATION
3.1 Modal Decoupling
Eq.(3) is solved here by separating the two variables x and t, and
using modal superposition:
V,(x,t) = Zi.,.2.,..<i>i(x)Vi(t) (13)
where (j); (x) is the ith mode shape and Vi(t) is the normal coordinate for that
1520
mode. Using modal orthogonality, Eq.(3) becomes a series of decoupled
equations of modal motion:
v,”(t) + 2 Q CO, V,>(t) + CO,' V,(t) = S, f’(t) (14)
where
cOi' - a^' El/m,
Ci = c,/(2 coim,)
Si == (1-x/L) dx / Io‘ (j); ^(x) dx (i = l,2...) (15)
(Oj and Q are respectively modal natural frequency and damping ratio. S; is a
factor indicating the extent of participation of the mode. The mode shapes
are given as
(l)i (x) = Cl i sin(aiX) + C2,i cos(aiX) + C3 ^ sinh(a|X) + C4 i cosh(aiX) (16)
a; related to the natural frequency is the solution to the eigen value problem,
and the coefficients of c, ■ to 04,; are obtained by meeting the boundary
condition in Eq.(5) for every mode.
3.2 Modal Solution in State Space
Let the state variables for mode i be denoted by
Vi(t) = (Vi(t). V,’(t), Vi’Tt))'^ (17)
where superscript T denotes transpose of matrix. The inclusion of Vi”(t) is
to provide a complete picture of the system behavior, and to be able to
describe the maxima of displacement as discussed later.
The solution to Eq.(14) can be shown in the following convolution
form:
Vi(t) = - Si C| 1 o‘ Zj (t-x) f’(x) dx (18)
where C; is a matrix of system parameters for mode i, and Zi(t) is a time-
dependent vector:
( 0
Ci = I 1
1/cOdi 0)
-Q cDi/cOdi 0 1 (19a)
(^i l^*
1521
(19b)
Zi(t) = (exp(- ©it) cos ©dit. exp(- ©jt) sin ©^jt, SCt))"^
where ©^j = ©i(l-^ and 5(t) is the Dirac delta function.
Since the ensemble expectation of Vi(t) is a zero vector due to zero
mean of the input process g(t), the covariance matrix of Vi(t) is
= E[Vi(t) v,^(t)] = S;^ Ci B|(t) C,^
Bi(t) = Jo'Jo'T|(T,)r|(-Cj)Rj5(|T,-tj|)Zi(t-T,)Zi'^(t-t2)<i'Ci(lt2 (20)
where E stands for ensemble expectation. This double integration has been
performed as shown in by analysis in the time domain.
It will be seen below that the covariance matrix for two different
modes (say m and n) is needed for further analysis. This matrix can be
similarly expressed as:
R.,,„(t) = EK(t) v^-^ft)] = S„ B,„(t) C J
Bmn(t) = lo%'^(^l)Tl('C2)RgE(Kr'r2l)Zm(t-'^l)ZnVT:2)dTidx^ (21)
This integration is also carried out in the time domain. The results of B^n(t)
are explicitly given in Appendix. Note that when m=n=i, B„n(t)= Bi(t).
Eqs.(20) and (21) show that the statistics of the nonstationary responses are
expressed as product of a stationary part (constant Cj matrix) and a
nonstationary part (matrix Bi(t) or B„n(t)).
3.3 Mean Square Total Response in State Space
With the elementary terms given, the covariance matrix Rvv(t) of the
vector of dynamics state-variables v(t) can be now formulated as follows:
v(x,t) = (V,(x,t), V,(x,t), V,”(x,t))’'
= (i:i=i,2....cl>.(x) V,(t), Zi=,2....<l)i(x)Vi’(t), i:i=,2..,.<^.(x)Vi”(t)) (22)
R,,(t) = E[v(x,t) v'^(x,t)]
= In.l.2,...^.n(x)S, (l)„(x)S„ C, B„,(t) Cj
f Rvv.ll(X,t) Rvv,,2(x,t) Rvv.l3(X,t) "l
= i R,v.2l(X,t) Rvv,22(X,t) Rvv.23(X,t) 1 (23)
^Rvv.3!(X,t) Rvv.32(X,t) R,,.33(X,t)j
Note that the second group of subscripts indicates the physical quantities by
1522
1= displacement, 2= velocity, and 3 = acceleration. This system of
identification will be used hereafter for variances. Of the summed terms in
Eq.(23), the dominant terms are often those referring to the same mode, i.e.
((^,(x)S„)^ CA(t)C7(n=l,2,...).
4. NUMERICAL EXAMPLES
4.1 Unit Step Modulation
The modulation function given in Eq.(7) is in a general form to
describe nonstationarity of earthquake excitations. It can be reduced to
several commonly used models. For example, the unit step function model
of modulation is a special case represented by N = l, ai = l, and bj=0 in
Eq.(7). This model is perhaps the earliest modulation envelope for modeling
nonstationarity of earthquakes When effects of the filter is eliminated by
setting (X)gi->oo and C0g2=0, the solution in Eq.(20) for a mode reduces to the
response of an oscillator to white noise modulated by the step function, with
natural frequency (Hq, damping and, of course, the modal participation
factor So = l in Eq.(14):
S exp(-2(;oCOot)
Rvvti(0= - {1+ - [ - l+^;o^cos2c0dot-(;o(l-^o^)‘^sin2co^ot]} (24)
4CoCio' (1-Co^)
S
Rvv p(t) = - exp(-2(;oC3ot)( 1 - cos2c0dot ) = Rvv,2i(f) (25)
4(Oo^(l-Co^)
S exp(-2i;oCOot)
Rvv 22(t) = - { 1 + - [ -1 +^o'cos2co,ot+^(l-Co')""sin2co,ot]} (26)
4(;o0)o
s exp(-2(:;otoot)
Rvv ,3(0 - - { -1 + - [l-2(:;o'+Co'cos2o3,ot+(;o(l<oO'''sin2o)dot]}
4^ot0o d-CoO
- Rvv.3,(0 (27)
S
Rvv 23(0 = - exp(-2(;oCOot)[ 1 + (l-2^o0cos2o3,ot -2^o(l- ^o0'^'sin2co,ot]
4(1-
1523
= Rvv.32(t)
(28)
Rvv.33(0 — ^ °° (29)
The first three terms have been given elsewhere The last term is
contributed by the white noise acceleration input, representing an impulse of
infinite peak at time lag t:=0.
4.2 Shinozuka-Sato Modulation
Another group of modulation models is described by the Shinozuka-
Sato model which is a special case of N==2 in Eq.(7). Examples are
a, =2.32, a2=-2.32, b,=-0.09, and b2=-1.49 for El Centro earthquake, and
a, = 12.8, a2=-12.8, bi=-0.14 , and b2=-0.19 for Taft earthquake The
latter is used in this example, with cOgj = 6.47: rad/s, = 0.65, C0g2 =
0.3271 rad/s, and = 0.5
For a typical reinforced concrete pier of highway bridges, L=12m,
EI/n\ = 250 kN/kg-m\ and ^,=0.04 (i=l,2,...) are used here. The constant
damping is used because it is felt that the one inversely proportional to the
modal frequency as defined in Eq.(15) is not realistic for concrete structures.
Three sets of stiffness ratio rt=ktLVEI and rf= k^L/EI are used here: q -> oo
and r,^ oo, q =1 and r=l, and r=0.1. The variances of displacement,
velocity, and acceleration at the top (free) end of the beam (x=L) are shown in
Figs.l to 3 for the three cases, respectively. Unit intensity is used here, coj is
the first natural frequency of the beam. It is seen that decrease of the
foundation stiffness causes the response to increase, showing the importance
to take into account the foundation stiffness. They also show that the peak
responses are slightly delayed by lower foundation stiffness. On the other
hand, additional analysis results show that the internal forces (shear and
moment) are reduced by lower foundation stiffness.
4.3 Application to Probabilistic Response Spectrum
The mean square response statistics shown here may be used to
develop probabilistic seismic response spectra for interested failure modes.
An example is given here. Consider a bridge span with one end supported
by an abutment with fixed bearings and the other by a pier with expansion
bearings. The fixed bearings and the bridge deck (superstructure) are
modeled as an SDOF system with mass M, damping C, and stiffness K
subjected to the strong motion acceleration f’(t) through the abutment:
Mw”(t) + Cw’(t) + Kw(t) = -Mf”(t) (30)
1524
where w(t) is the horizontal displacment of the deck relative to the abument.
The relative displacment u(t) at the end supported by the pier is focused
here:
u(t)= w(t)- V(L,t) (31)
where u(t)=(u(t),u’(t),u”(t))'^. It is critical to understand the dynamics of
the system when designing the bridge to cover the failure mode of span
collapse due to excessive relative displacement u(t). The covariance of u(t)
can be readily expressed as
^uu(0 — Rww(0 " ^wv(0 ■ Rvw(0 Rw(l)
Rvv(t) has been given in Eq.(23), R^,,(t) can be expressed using Eq.(20),
R,^v(0 and Rv„.(t) can be readily formulated using Eq.(21) and taking
summation over the significant modes involved in vector V(t). Note that the
length of cantilever beam L in Eq.(31) is omitted in Eq.(32).
Now let us define the failure of excessive relative displacement. Let
E[M(-co,t)] be the mean rate of displacement maxima at time t and
E[M(U,t)] be the mean rate of displacement maxima above a given level U,
they are respectively
E[M(-oo,t)]= - L"duL° u” p,,. „>(u,0,u”,t)du” (33)
E[M(U,t)] = - Iu"duL° u” „..(u,0,u”,t)du” (34)
where p^ ,,- ^•■(.) is the joint probability density function for u, u’, and u”. If
g(t) in Eq.(6) is assumed to be a Gaussian excitation, these state variables
are also Gaussian variables. E[M(-co,t)] and E[M(U,t)] are then derived as:
?^'(t) Ruu.33'^^(t) 1
E[M(-oo,t)] = - (35)
27: K..22%) 1“X'(0
U
E[M(U,t) = E[M(-«5,t)] {0( - )
Mt) Ruu.ii'^'(t)
U'
- x(t) exp[ - (l-x^(t))] 0(-
2X^(t) R,,H(t)
X(t)U
- ) } (36)
X(t) R,,,n^^^(t)
where
1525
|R(t)l
X^(t) = -
^u,ll(0 ^u.22(0 ^u,33(0
X(t) = y,3(t) - Yl2(t) Y23(t)
Ymn(t) = Ruu.mn(t)/[I^u.rnm(t)R^u.nn(0]’^ (m,n= 1 ,2,3 , HI II) (37)
|R(t)| is the determinant of R„„(t), and 0(.) is the cumulative probability
function for the standard normal variable.
Let probability of failure due to excessive displacement be
Pf = Probability [ maximum u > U ]
= E[M(U,t)] dt / E[M(-oo,t)] dt (38)
where Tj is the interested time length, being the time interval of significant
seismic input. Pj- indicates the likelihood that peak displacements exceed a
given level U. For a given Pf and seismic input intensity S, variation of the
threshold level U as a function of structural system frequency and damping
is defined here as probabilistic displacement response spectrum, because it is
associated with a probability to be exceeded. This spectrum can be used for
design to control displacement. For example, the minimum seat width
requirement for highway bridges subjected to seismic hazard may be derived
using this approach for risk-based design. For the example of bridge pier
discussed above, it is found that when the pier’s first natural frequency is
close to that of the superstructure-bearing system, U may be significantly
lower. This is because the two systems behave very much similarly due to
their similar dynamic properties so that their relative displacement u(t)
approaches zero. Note that this concept of probabilistic response spectra can
be applied to acceleration for force control design.
5. CONCLUSIONS
An explicit solution is presented in this paper for random vibration of
elastically supported cantilever beams subjected to the strong ground motion
modeled by modulated cascade-filtered white noise. It may be used to
develop probabilistic response spectra for risk-based seismic design and
retrofit specifications.
REFERENCES
[1] G.Ahmadi and M.A.Satter "Mean-Square Response of Beams to
Nonstationary Random Excitation", AIAA Journal, Vol.l3, No.8, pp.l097-
1526
1100
[2] C.G.Bucher "Approximate Nonstationary Random Vibration Analysis
for MDOF Systems", J.App.Mech. Vol.55, pp.197-200, 1988
[3] T.K.Caughey and H.F.Stumpf "Transit Response of A Dynamic System
under Random Excitation, J. Appl. Mech., ASME, Vol.28, pp. 563-566,
1961
[4] R.W. Clough and J.Penzien. Dynamics of Structures, McGraw-Hill
Book Company, 1975
[5] R.B.Corotis and T. A. Marshall. "Oscillator Response to Modulated
Random Excitation, " ASCE J.Eng.Mech. Vol.103, EM4 pp. 501-5 13, 1977
[6] G.Fu "Seismic Response Statistics of SDOF System to Exponentially
Modulated Coloured Input: An Explicit Solution” Earthquake Engineering
and Strucmral Dynamics, Voi.24, 1995, pp. 1355-1370
[7] D.A.Gasparini "Response of MDOF Systems to Nonstationary Random
Excitation", ASCE J. of Engg. Mech., Vol.105, No. EMI, Feb. 1979,
pp. 13-27
[8] D.A.Gasparini and A.DebChaudhury "Dynamic Response to
Nonstationary Nonwhite Excitation" ASCE J.Eng.Mech. Vol.106, EM6,
pp. 1233-1248, 1980
[9] Z.K.Hou "Nonstationary Response of Structures and Its Application to
Earthquake Engineering", California Institute of Technology, EERL 90-01,
1990
[10] W.D.Iwan and Z.K.Hou "Explicit Solution for the Response of Simple
Systems Subjected to Nonstationary Random Excitation", Structural Safety,
Vol.6, 1989, pp.77-86
[11] K. Kanai "Semi-empirical Formula for the Seismic Characteristics of
the Ground", Univ. of Tokyo Bull. Earthquake Res. Inst., vol. 35, pp. 309-
325, 1957
[12] Y.K.Lin. Probabilistic Theory of Structural Dynamics, R.E.Krieger
Publishing Company, 1967
[13] M.Shinozuka and Y.Sato "Simulation of Nonstationary Random
Process", ASCE J.Eng.Mech. Div., Vol.93, pp.11-40, 1967
[14] M.Shinozuka and W-F.Wu "On the First Passage Problem and Its
Application to Earthquake Engineering", Proc. 9th WCEE, Aug. 2-9, 1988,
Tokyo-Kyoto, Japan, P.VIII-767
[15] H.Tajimi "A Statistical Method of Determining the Maximum Response
of a Building Structure during an Earthquake", Proc. 2nd World Conf.
Earthquake Eng. Tokyo and Kyoto, vol.II, pp.781-797, July 1960.
[16] C.-H. Yeh and Y.K.Wen "Modeling of Nonstationary Ground Motion
and Analysis of Inelastic Structural Response" Structural Safety, Vol. 8,
1990, pp.281-298
Appendix
1527
(Al)
f ®mn.ll ^mnJ2 ®mn,13
®mn(0 1 ®mn,2I ®mn,22 ®mn,23
^ ®mn,3I ®mn,32 ®mn.33
B.n.n(t) = S eXp(-0)„^„-C0„(;n)t - 2i.,.N Ij=,.N aiHj *
^^^glc®gk
{(Cak + c J [g 1 (akn„a,„, Pj,„, ^i,„)-g2(ai^, p kniPjlansP-ikn)
§3(^kiti> ^kn» Pjkm » Pikn)“§4(^ km ’ Pkn » Pjkm » Pikn)
S 1 ( P km ’ Pkn > Pjknn Pikn) ”§2 (Pkm > ^kn » Pjkm 5 Pikn)
§3 ( P km » Pkn » Pjkm > Pikn) ~S4( Pkm ’ ^kn* Pjkm ’ Pikn)
S5(^km5 ^ktn Pjkm» Pikn)“S6(^km’ Pkm Pjkm’ Pikn)
+ g7(akm>a
kn»Pjkm> Pikn)~S8(^km» Pkn’Pjkm’ Pikn)
*^g5(Pkm’Pkn»Pjkm>Pikn)“g6(Pkm>^kn»Pjkm»Pikn)
+ g7(Pkm>Pkn»Pjkm,Pikn)-g8(P km » ^kn ’ Pjkm » Pikn)
~^Cgk®gk^®gdk(^ak”^bk) [g9(^km > ^kn » Pj km > Pikn) ~glo(^km>Pkn>Pjkm’ Pikn)
+gii(akm>a kn 5 Pjkm > Pikn)‘g 1 2 (*^km > P kn » Pjkm ’ Pikn)
~g9(Pkra’Pkn»Pjkm’Pikn)”^glo(Pkm’^kn’ Pjkm ’ Pikn)
"gil(Pkm>Pkn’Pjkm>Pikn)"^gl2(Pkm’^kn» Pjkm » Pikn)
gl sC^km ’ ^kn» Pjkm ’ P ikn)”g 14(^km > Pkn > Pjkm » Pikn)
“^gl5(*^km>*^kn>Pjkm»Pikn)“gl6(^km»Pkn’PjkmJpikn)
“glS^Pkm’Pkn’Pjkm’Pikn) "^gl4(Pkm ’*^kn’ Pjkm’ Pikn)
“gl5(Pkm’Pkn’Pjkra’Pikn) gl6(Pkm’^kn’ Pjkm’ Pikn)
Bmn.l2(0 — Bnm.2l(0
1
B„,n.i3(0 = S [Zi=,,,s, a, exp(bit)] I, =1.2 - aj exp(bjt) *
S^gk^gk
{(^ak“^^bk)[gl8(^ km’ Pjkm) '^gl8(P km’ Pjkm)]
■*"^gk®gk^®gdk(Cak"^bk)["gl7(^km’Pjkm)"*"gl7(Pkm’Pjkm)]}
1
B„n,2i(t) = S exp(-®^i;„-co„Qt I, ^, 2 - 2^i=i.N Ij=i,N Siaj
16Cgk<«gk
{ (^ak ^bk) ["g9(^km ’ ^kn ’ Pjkm ’ Pikn) g 1 o(^km ’ P kn ’ Pjkm ’ Pikn)
~g 1 1 (^km ’^kn ’ Pjkm ’ Pikn) g 1 2 (®'km ’ P kn ’ Pjkm ’ Pikn)
”g9(Pkm ’ Pkn» Pjkm’ Pikn) g lo(Pkm’^kn’ Pjkm ’ Pikn)
~gl l(Pkm’Pkn ’Pjkm’ Pikn) '^g|2(P km’ ^kn’ Pjkm ’Pikn)
"b g 1 3 (Ct itm ’ ^kn ’ Pjkm ’ P ikn)“g 1 4(^km ’ Pkn ’ Pjkm ’ Pi kn)
“b g t5(^km ’^kn’ Pjkm’ Pikn)“gl6(^km’ Pkn’ Pjkm’ Pikn)
+ gl3(Pkm’P kn’ Pjkm’ Pikn) gl4(Pkm’^kn’ Pjkm’ Pikn)
(A2)
(A3)
(A4)
1528
"^SlsCPkm’ Pkn»!^jkm>Pikn)”§16(P km ’ *^kn ’ P'jkm » P ikn)
^gk® gk^ ® gdk (^ak”^bk) [Sl(^km’^kn’Pjkm» M'ikn)~S2(*^km >Pkn’Pjkm»Pikn)
§3(*^km5^kni Pjkm’ Pikr)“S4(^km» Pkn’ Pjkm’P'ikn)
“§ 1 ( Pkm ’ Pkn > Pj km ’ M-ikn) S2(P km » *^kn > Pjkm » M-ikn)
"SsCP km > Pkn ’ Pjkm » M-ikn) §4(P km ’ ^kn * Pjkm > l^ikn)
"S5(*^km5 ^kfijM-Jlcm’ Pikn) §6(^km> Pktu M-jkm’ Pikn)
-g7(ak^,a;,„,^jkm.Pikn)+g8(Ct km > Pkn > M’jkm ’ Pikn)
g5(Pkm’ Pku’ M-Jkm* Pikn)~S6(Pkm»^kn> Pjkm> Pikn)
S7(Pkm » Pktu Pjkm » Pikn)"g8(Pkm > *^kn? Pjkm» Pikn) (-^5)
1
®mn,22(0 ~ ^ eXp(-COn,!l^n;,-CDj,(!^j,)t 2 ^i = l,N ^j = l,N ^i^j
16Cgk®gk
{ (^ak ^bk) [g 1 (^km »^kn’Pjkm’Pikn)'^g2(^km>Pkn’Pjkm»Pikn)
g3(*^km ’ ^kn ’ Pjkm > Pikn) g4(*^km > Pkn J Pjkm ’ Pikn)
”^gl(Pkm»Pkn’Pjkm»Plkn)”^g2(Pkm’^kn»Pjkm5pikn)
g3(Pkm» Pkn» Pjkm’Pikn) g4(Pkm»*^ knJpjkm>Pikn)
+ g5(Ctkm>CX,„,^jkm.Pikn) + g6(a km’Pkn»Pjkm>Pikn)
g7(^km > ’^kn » Pjkm ? Pikn) g8(^km > Pkn » Pjkm ’ Pikn)
“^gsCPkm’ Pkn»Pjkm>Pikn) "^ge^P km » ^kn ’ Pjkm 5 P ikn)
■^g7(Pkm’Pkn'! Pjkm’ Pikn) “^g8(Pkm»^kn>Pjkm5pikn)]
^gk^gk^^gdk(^ak~^bk) [g9(^km’*^kn’ Pjkm ’ Pikn) glo(^km’ Pkn’ Pjkm? Pikn)
g n (^km ’ ^kn ’ Pjkm ’ Pikn) g I2(^km ’ Pkn ’ Pjkm » Pikn)
■g9(Pkm>Pkn’Pjkm>Pikn)"glo(Pkm»^kn’Pjkm’Pikn)
"gH(Pkm’Pkn’Pjkm’Pikn)“gl2(Pkm’^kn’ Pjkm’ Pikn)
+ gl3(akm’akn’Pjkm’Pikn) + gl4(akm’ Pkn’ Pjkm’ Pikn)
gl5(^km’*^kn’ Pjkm’ Pikn) "^gieC^km’ Pkn’ Pjkm’ Pikn)
"gl3(Pkm’Pkn»Pjkm’Pikn)"gl4(Pkm’^kn’ Pjkm’ Pikn)
■g IsCPkm ’ Pkn’ Pjkm ’ Pikn)“g leCPkm’^kn’Pjkm’ Pikn)]
(A6)
1
^mn,23(0 ~ S [I]j=i,N ^i ®^P(bit)] I]ic = l,2 Sj = l.N ^j ®XP(t>jt)
^^gk®gk
{ (^ak ^bk) [gl7(^km ’ Pjkm) gl7(Pkm > P jkm)]
*^gk® gk^^gdk(^ak“^bk) [gl S^^^km ’ Pjk)"gl sCPkm ’ Pjkm)! }
(A7)
Smn,3l(0 = Bnn, 13(1)
(A8)
®mn.32(0 Bnm.lsCO
(A9)
B^n.33(t) = E=i.n a; exp(bit)]' Rgg(O)
(AlO)
1529
Vnriance
where P,,. and p,„ are totions of system parameters givea below;
CL^ = - CiDjdk s • Pier - ^dn “ ’
Pjerr = bj “b ’ P-ilcn ”
(ij = l,2 . N; k=l,2; m,n=l,2,...)
Note chat fanctions g, through gt8 are
-h - Cj,C0j,
(All)
not siven here due to limited space.
Fig.l Variance of Top Displacement, velocity, and Acceleration
(S = 1 , r, -» and ®)
1532
LINEAR MULTI-STAGE SYNTHESIS
OF RANDOM VIBRATION SIGNALS
FROM PARTIAL COVARIANCE INFORMATION
S.D. Fassois* and K. Denoyer**
*Deparl-,menl'. of Mechanical & Aeronautics Engineering
University of Patras
GR-265 00 Patras, Greece
E-mail: fassois@mech.upatras.gr
**Department of Mechanical Engineering & Applied Mechanics
The University of Michigan
Ann Arbor, MI 48109-2125, U.S.A.
ABSTRACT
An effective scheme for random vibration signal synthesis from partial co-
variance information is introduced. The proposed scheme is based upon
a Fast Rational Model estimation approach, combined with a discrete
ARMA(2n,2n-l) vibration signal representation and a dispersion analysis
methodology. Unlike previous approaches, the proposed scheme provides
accurate synthesis without resorting on nonlinear operations, and provides
the tools for effective representation order determination. Rirthermore, it
mathematically guarantees the estimated representation stability and in-
vertibility properties, implying that all types of vibration signals, including
those characterized by sharp spectral peaks or valleys, can be handled with¬
out difficulty. The effectiveness of the scheme is demonstrated via synthesis
test cases, through which the benefits of representation overfitting are also
presented.
G{s)
h
S[u)
7(t)
7[/cri
4>[B)
6{B)
4>i
Oi
t-H
List of Symbols
receptance transfer matrix
j-th term of the Green’s function
j-th term of the inverse function
power spectral density function
continuous time auto covariance function
discrete time auto covariance function
autoregressive polynomial
moving average polynomial
i-th autoregressive parameter (i = 1, 2, • • ■ , 2n)
i-th moving average parameter {i = 1, 2, • • * , 2n - 1)
■i-th continuous pole (i = 1, 2, • • • , 2n)
1533
5i ; z-th vibration mode dispersion percentage (i = 1, 2, • • • ,n)
cr^ : innovations variance
B : backshift operator (J5rr[i] = — 1])
R : number of specified autocovariance samples
T : sampling period
p : truncated inverse function order
no : number of structural degrees of freedom
n ; number of degrees of freedom in the signal representation
Superscript:
^ : indicates iteration number.
Conventions:
[•] : function of a discrete variable.
(•) : function of a continuous variable.
Vector/matrix quantities are indicated in bold-face characters.
1. INTRODUCTION
In this paper the problem of digitally synthesizing stationary random
vibration signals from partial (incomplete) covariance information is con¬
sidered. The problem is important for generating vibration signals to be
used in the computer or laboratory based simulation and testing of various
types of structural systems, including machine, vehicle, aerospace, and civil
engineering structures.
The problem of equi-spaced in time vibration signal synthesis from
complete, analytically available, covariance information may be tackled via
well known spectral factorization techniques [1], which yield an appropriate
stochastic realization that may be subsequently used for synthesis purposes.
The practically much more important problem of synthesis based upon a
finite (truncated) number of available covariance samples is equivalent to
that of finding an admissible extension of the partially specified covariance
sequence, and obtaining an analytical description that can be used for
vibration synthesis. A complete analytic description, which automatically
implies an admissible extension of a given covariance sequence, as long as
the latter corresponds to a rational spectral density, is provided through
the AutoRegressive Moving Average (ARMA) representation [2].
Limiting attention to purely AutoRegressive (AR) based vibration
synthesis leads to the well-known Yule-Walker equations which are linear in
terms of the representation parameters [2]. Unfortunately purely AR based
schemes are ineffective for proper synthesis, especially when the partially
specified covariance corresponds to relatively sharp valleys in the spectral
domain [3,4]. Purely Moving Average (MA) representations, such as the
ones advocated by Abdul-Sada and Mahmood [.5] in a different context,
are also inappropriate for describing the relatively sharp spectral peaks
1534
of vibration signals. These considerations unambiguously lead to mixed
Autoregressive Moving Average (ARM A) based vibration signal synthesis.
Unfortunately, ARM A l-)ased synthesis is a much less tractable prob¬
lem, leading to a non-linear mathematical optimization problem [1-3]. To
circumvent this difficulty Gei'sch and Luo [6] proposed a mixed scheme, in
which the AR parameters are detained through the Yule- Walker equations,
while the MA parameters are obtained via a non-linear modified Newton-
Raphson procedure. In subsequent work, Gei'sch and co-workers [7,8] uti¬
lized proper extensions of the Two Stage Least Squares (2SLS) method
introduced in time series analysis by Durbin [9]. The 2SLS method is, nev¬
ertheless, known to be characterized by statistical inefficiency and limited
achievable accuracy [10]. In related work within a broader context, Geor-
gioii [11] attempted finding an admissible extension of a partially specified
covariance sequence via a method that recursively updates ARMA repre¬
sentations of dimension increasing with the sequence length. This method,
however, requires a-priori information on the representation’s zero loca¬
tions. For very simple cases, corresponding to low order MA polynomials,
this information may be, perhaps, derived from the asymptotic behavior
of the partial autocorrelation coefficients. Nevertheless, such a procedure
cannot be used with realistic MA polynomials of interest in the vibration
synthesis problem.
In this paper an effective scheme for ARMA based vibration signal
synthesis is introduced. The scheme is based upon a Fast Rational Model
estimation approach [3,4], combined with the covariance invariance prin¬
ciple and a dispersion analysis methodology. The covariance invariance
principle [2] is used for relating the continuous and discrete vibration sig¬
nal representations, and leads to a special ARMA(2n,2n-l) structure for the
latter. The Fast Rational Model approach, originally introduced within the
context of spectral estimation from available signal samples [3,4], is used
for effective parameter estimation, while the dispersion analysis method¬
ology [12] is used as the main tool for representation order determination.
The scheme uses the specified autocovariance samples as a pseudo-sufficient
statistic, and develops a strongly consistent ARMA representation based
upon exclusively linear operations. An additional important property of the
proposed scheme is that it is capable of mathematically guaranteeing the
stability and invertibility of the obtained ARMA representation; a feature
implying that, unlike alternative techniques, it can handle the practically
significant classes of vibration signals characterized by sharp spectral peaks
or valleys without any difficulty.
The paper is organized as follows: The proposed vibration signal
synthesis methodology is presented in Section 2, synthesis test cases are
considered in Section 3, and the conclusions of the work are summarized
in Section 4.
2. VIBRATION SIGNAL SYNTHESIS METHODOLOGY
2.1 Fundamentals
Consider a linear viscously-damped rio degree-of-freedom structural
1535
system described Id.y the vector differential equation:
M • x(t) -I- C • x(i) -f- K ■ x(0 = f{i) (1)
where M,C,K represent the Uo x Uo mass, viscous damping, and stiffness
matrices, respective!}^, and f(^), x(i) the ?7.o-dimensional force excitation
and vibration displacement signals, respectively. Laplace transforming (1)
leads to:
X(s) = [M • .5- -I- C • s -I- K]-‘ • F(.s) = G(s) • F(s) (2)
in which s represents the Laplace Transform variable and G(s) the system’s
receptance transfer matrix.
Let Gij[s) represent the ij-th element of G(s), that is the transfer
function relating the Lth displacement with the j-th force. Assuming that
the force excitation is a stationary zero-mean imcorrelated stochastic signal,
that is:
= 0 • fjito)} = aj ■ 6{ti - £2)
with E{-} indicating statistical expectation and 5(-) the Dirac delta func¬
tion, the resulting vibration displacement Xi{t) will, in the steady-state, be
a stationary stochastic signal with zero mean and autocovariance [2]:
roo
7(r)=crP/ + = (3)
k=l
In these expressions (the latter of which assumes distinct poles) g{t) repre¬
sents the impulse response of the receptance transfer function Gij{s), and
dk {k = 1,2, •• • ,2no) the structural poles and corresponding autoco¬
variance expansion coefficients, respectively.
The uniform instantaneous sampling of the stochastic vibration dis¬
placement x{t) (the subscript being, for the sake of simplicity, suppressed)
leads to a discrete-time stochastic signal x[kT] {k = 0,1,2,-’*), with T rep¬
resenting the sampling period, which is characterized by an autocovariance
function 'y[kT] satisfying the covariance invariance principle [2]:
7[/:Tl=7('r)|r=/,-T (4)
Assuming that ,t(£) is band-limited, and that the conditions of the Nyquist
theorem are satisfied, the continuous and discrete spectral densities satisfy
the relationship [1]:
S°(a.) = is7a,) (_!<„< 1) (5)
in which oj represents frequency in rads /sec, and S^{u) the con¬
tinuous and discrete densities, respectively.
Denoting the discrete time kT in normalized form as simply k, or
£ (£ = 0,1,2,---), the sampled displacement signal may be shown [2] to
1536
admit a stable and invertible AutoRegressive Moving Average (ARM A)
representation of orders (2no,2?ro — 1), that is:
2710 2710-1
x\t] f ^ (f)k • x[t - /c] = ry[tl -!- ^ Oj, • w[t - /c] ==^
A:=l k=l
=> • x[t] = f){B) ■ rult] (6)
In the above iv\t] represents a discrete, zero-mean, imcorrelated (inno¬
vations) sequence with variance and (f>ic {k = 1, 2, ■ • - , 27io) and 9k
(/c — 1,2, • • • , 271o - 1) the autoregressive (AR) and Moving Average (MA)
parameters, respectively. The AR and MA polynomials are defined as:
<t>(B) = + 3{B) =
with B denoting the backshift operator (S • x[t] = a;[t - 1]).
Once determined from the specified partial covariance sequence, a
proper ARM A I'epresentation of the form (6) may be used for vibration
signal synthesis.
2.2 The Algorithm
The vibration synthesis problem treated concerns the generation (re¬
alization) of discrete random vibration displacement signals with covariance
characteristics conforming to a prescribed partial (truncated) autocovari¬
ance sequence ^[k] (/c ~ 0, 1, 2, • - ■ , i?).
The vibration signals of interest may be viewed as responses of linear
viscously damped structural systems of the form (1) subject to uncorre¬
lated force excitation, and may be synthesized via discrete ARMA(2n,2n-l)
representations of the form (6) driven by realizations of pseudo-random un¬
correlated sequences characterized by zero mean and variance cr,^. Toward
this end iDoth the ARM A representation and innovations varipce cr^ need
to be determined from the given covariance information. This is achieved
via the Fast Rational Model estimation approach, which consists of a se¬
quence of stages, with each one based on exclusively linear operations [3,4],
as well as the dispersion analysis methodology (see [12] for a treatment
of the notion of modal dispersion) for representation order determination.
The resulting synthesis algorithm may be outlined as follows:
In Stage 1 a truncated, p-th order, version of the ARMA represen¬
tation’s inverse function {IjYj^i {h = 1) [R > V > 2?! with p generally
selected -cis p ^ (2.5 - 5) x 2n] is obtained through the estimator expres¬
sions;
j2lj-l[k-j]+l[k] = 0 (/c = l,2.--.,p) (7)
In Stage 2 initial MA parameter estimates 9°. {k = 1, 2, • • • , 272 — 1)
1537
are obtained by solving the set of equations:
2n-l
= (i = 2n-|-l,---,4n-l) (8)
A:=l
□
In Stage 3 AR parameter estimates d)] (j = 1, 2, ■ ■ • , 2n) are obtained
via the estimatoj* expressions:
= 0 (/c = l,2,---.2n) (9)
in which the superscript i denotes iteration number and 7^ ^[fc] the auto¬
covariance function of the signal:
1*1
a:[t]
(10)
The samples of this autocovariance are computed by assuming
7*"' [A] ss 0 for 1*1 > R. This leads to the following set of equations,
which may be solved for q'-'lfc] (i: = 0, 1, • • • , R):
X:A(i,;)-y-1i] = 7b1
1=0
in which:
aj^i I = 0
aj+i -b I ^ 0
where aj = 0 for j 0 (0, 2n - 1], and:
2n~l~j
k=0
(11)
(12)
(13)
□
In Stage 4 updated MA parameter estimates 0]- (j = 1, 2, ■ • • , 2n - 1)
are obtained based upon the expressions:
9} = - E 4 • (i = 1, 2, • • ■ , 2n - 1) (14)
In Stage 5 the innovations variance is estimated as:
(4)^ =
T[0] + Ei*b'l'[l1
j=l
2n-l
1-H
n -1
(15)
1538
with Cj representing an estimate, obtained at iteration i, of the j-th sam¬
ple of the AR.MA representation’s Green’s function [2]. This is achieved
through the recursive expressions:
G; = «'-X;4-C% (j=^l,2,---,2n-l) (16)
A:=l
with = 1, G) ^ 0 for j < 0. Following this, the representation’s
aiitoco variance generating function and spectral density may be obtained
as:
fiB) ^ f: 7’ifcl-B" =
k=-oo
0^{b)-0Hb-^) o ,
with j denoting the imaginary unit.
In Stage 6 the dispersion percentage <5^ [j - 1, 2, • • - , n) of each vi¬
bration mode, expressing the mode’s normalized energy contribution in the
signal representation, is (in the underdamped case) computed as:
= (%) {j = l,2,---,n) (18)
with the quantities dj (j = 1, 2, • • • , 2?^) being defined by Eq. (3). □
Stages 1 through 6 are repeated for signal representations with suc¬
cessively increasing number of degrees of freedom n, and, in Stage 7, the
proper number of degrees of freedom is selected as:
n* = max{ n | 15j| > e Vj} (19)
with e indicating a small positive threshold value selected in accordance
with the desired synthesis accuracy. O
In Stage 8 the vibration signal is synthesized by exciting the obtained
ARMA(27^*,27^'‘ - 1) representation by a simulated zero mean uncorrelated
sequence with variance ((7,^y)\ C
Remark 1. For a given signal representation order, stages 3 and 4 of the
algorithm are iterated until convergence is achieved. The estimation results
of the iteration characterized by minimal innovations variance (cr.^)^ are
selected as best.
Remark 2. The invertibility of an estimated ARMA{2n,2n-l) representation
may be mathematically guaranteed via slightly modified MA parameter es¬
timators in stages 2 and 4, that are based upon the stability property of
zero-lag least-squares inverses (alternative versions A and B in Fassois [4]).
Alternative version B may be used in stage 3 in order to guarantee the
representation stability as well. The guaranteed stability and invertibility
properties are of primary importance, as they imply that all types of vi¬
bration signals, including those characterized by sharp spectral peaks or
valleys, may be synthesized without difficulty (see section 3).
1539
mi = 0.5 kg
ki = 100 N/m
Cl = 0.5 N*sec/m
m2 = 2.0 kg
0
0
rH
II
C2 = 0.5 N*sec/m
m3 = 1.0 kg
*3 = 150 N/m
C3 = 1.5 N*sec/m
*4 = 100 N/m
C4 = 1.5 N*sec/m
cs = 0.6 N*sec/m
C6 = 0.5 N*sec/m
Figure 1: Schematic representation of the three degree-of-freedom struc¬
tural system of Test Case I.
3. SYNTHESIS TEST CASES
Two test cases, in which the scheme is used for the synthesis of sig¬
nals matching a specified portion of the covariance structure of selected
vibration displacements, are presented.
3.1 Test Case I
In Test Case I the synthesis of signals matching part of the covari¬
ance structure of the displacement of mass 1 of the three degree-of-freedom
system of Figure 1 when excited at that point by a continuous-time uncor¬
related force is considered. This type of signal is characterized by spectral
peaks at 1.04 and 3.32 Hz^ with a spectral valley between them, as well as
a much less obvious third peak at 2.68 Hz (solid curve in Figure 2).
The auto covariance function jlk] is specified at lags k ~ 0, 1,2, • • •,
50 {R = 50), the sampling period is selected as T = 0.075 secs, the inverse
function order as p = 25, and the maximum allowable number of iterations
imax — 25.
The spectral density of the synthesized, theoretically sufficient, ARMA
(6,5) representation is compared to the theoretical spectral density in Fig¬
ure 2(a), from which very good agreement is observed. A similar remark
may be made for the autocovariance function of the synthesized represen¬
tation (not shown).
Allowing the I'epresentation order to be increased to 2n = 8 [ARMA
1540
POWER SPECTRUM (dB)
Figure 2: Synthesized ( - ) versus theoretical ( — ) spectral density func¬
tion: (a) ARM A (6,5) based synthesis, (b) ARM A (8,7) based synthesis
(Test Case I; fi = 50).
POWER SPECTRUM (dB)
O
z
-1.0
O ~ SPECIFIED
X - SYNTHESIZED
1 0
LAG
(a)
20
(b)
Figure 3; ARM A( 10,9) based synthesis results: (a) synthesized (x) versus
specified (o) auto covariance function, (b) synthesized ( - ) versus theoret¬
ical ( — ) spectral density function (Test Case 1] R = 50).
3 0
TIME (sec)
Figure 4: ARMA{10,9) based vibration displacement signal realization
(Test Case 1; R = 50).
(8,7) model] leads to visible improvements in the synthesized representa¬
tion spectral density [Figure 2(b)], as well as in its autocovariance. These
improvements are more evident at autocovariance lags higher than 10, and
at frequencies corresponding to the neighborhood of the spectral valley
located between the two peaks.
A further increase of the representation order to 2n = 10 [ARMA(10,9)
model] leads to an almost perfect match of both the specified autocovari¬
ance and spectral density, as the theoretical and synthesized curves are
practically indistinguishable (Figure 3). The extra poles introduced in this
case may be grouped into two pairs: a real and a complex conjugate, and are
both characterized by small dispersion percentages. These results suggest
that, despite the theoretical sufficiency of the ARMA(6,5) representation,
which is also confirmed by the Akaike Information Criterion (AIC) [1], over¬
fitting is beneficial in achieving the highest accuracy in matching the spec¬
ified auto covariance characteristics. Portion of a vibration displacement
signal realization obtained by the synthesized ARMA(10,9) representation
is depicted in Figure 4.
The effects of the number of lags R, at which the autocovariance
function is specified, have been also investigated. Lowering R to 35 (from
its current value of 50) does not appreciably change the synthesized repre¬
sentation’s spectral density, although it does lead to slight deterioration in
the neighborhood of the spectral valley and at the high frequency end. On
the other hand, increasing R to 100 produces a spectral density that is es¬
sentially identical to that of the R = 50 case, and almost indistinguishable
from the theoretical.
3.2 Test Case II
In Test Case II the synthesis of signals matching part of the covari-
1543
mi = 1,0
kg
II
0
0
N/m
m2 = 1-0
kg
0
0
II
N/m
m3 = 2.0
kg
0
0
II
N/m
0
0
r-t
II
►3?
N/m
Cl = 0.6
N*sec/in
C2 = 0.5
N*sec/m
C3 = 0.6
N*sec/m
C4 = 1.5
N*sec/m
C5 = 0.5
N*sec/m
C6 = 0.7
N*sec/m
Figure 5: Schematic representation of the three degree-of-freedom struc¬
tural system of Test Case II.
ance structure of the displacement of mass 1 of the three degree-of-ffeedom
system of Figure 5 when excited at that point by a continuous- time un¬
correlated force is considered. This type of signal is characterized by three
spectral peaks at 1.29, 2.03 and 2.81 Hz, with sharp valleys among them
[solid curve in Figure 6(b)] - a case of recognized difficulty.
The aiitoco variance function 'y[k] is specified at lags k — 0,1,2, •••,
50 {R = 50), the sampling period is selected as T = 0.0884 secs, the inverse
function order as p — 25, and the maximum allowable number of iterations
3-3 i-max ~ 25.
Similarly to the previous case, vibration signal synthesis is based upon
an overfitted ARM A (10, 9) representation. The extra poles include a real
and a complex conjugate pair, and are all characterized by small dispersion
percentages. The auto covariance and spectral density of the synthesized
ARMA(10,9) representation are compared to those specified in Figures 6(a)
and 6(b), respectively. Evidently, the autocovariance function is very close
to the specified, and certainly very satisfactory, although the matching is
not characterized by the “perfection” of the previous case. This is verified
from the representation’s spectral density plot as well, which indicates some
deviation from the theoretical density, especially in the neighborhood of the
first - and sharpest - spectral valley. This is, nevertheless, expected, due
to the difficulties associated with signals characterized by sharp spectral
valleys. The fact itself that the algorithm operated properly in this case is
due to its important inherent stability characteristics. Portion of a vibra¬
tion displacement signal realization obtained the estimated ARMA(10,9)
1544
(a)
(b)
Figure 6: ARM A( 10,9) based synthesis results: (a) synthesized (x) versus
specified (o) auto covariance function, (b) synthesized ( - ) versus theoret¬
ical ( — ) spectral density function (Test Case II; R = 50).
Figure 7; ARM A (10,9) based vibration displacement signal realization
(Test Case II: R — 50).
representation is presented in Figure 7.
4. CONCLUDING REMARKS
In this paper a linear multi-stage scheme for effective random vibra¬
tion signal synthesis from partial covariance information was introduced.
The scheme was shown to achieve accurate synthesis without resorting on
nonlinear operations. Moreover, by providing mathematically guaranteed
estimated ARM A representation stability and invertibility, it was shown
to be capable of effectively synthesizing all types of vibration signals, in¬
cluding the difficult classes of signals characterized by sharp spectral peaks
and/or valleys. In particular:
• Signals characterized by sharp spectral valleys were found to be most
difficult to synthesize, with the synthesis accuracy generally lagging
in the neighl>orhood of the spectral valley.
• Representation overfitting, with the additional modes characterized
by relatively small dispersion percentages, was shown to be necessary
in achieving maximum synthesis accuracy.
• The required minimal number of autocovariance samples for accurate
synthesis was found to be about 12no, with no denoting the number of
structural degrees of freedom. Increasing the proportionality factor
to about 16 was shown to lead to some improvement, but further
increases were of no significance.
1546
REFERENCES
[1] SM. Modern Spectral Estimation: The.ory and Application. Pre¬
ntice-Hall, 1988.
[2] Pandit S.M. and Wu, S.M. Time Series and System Analysis vjith
Applications. John Wiley and Sons, 1983.
[3] Fassois, S.D. A fast rational model approach to parametric spectral
estimation. Part I: the algorithm. ASME Journal of Vibration and
Acoustics, Vol. 112, 1990, pp. 321-327.
[4] Fassois, S.D. A fast rational model approach to parametric spectral
estimation. Part II: properties and performance evaluation. ASME
Journal of Vibration and Acoustics, Vol. 112, 1990, pp. 328-336.
[5] Abdul-Sada, J.W. and Mahmood, M.K. Generation of gaussian pseudo¬
random process with specific correlation properties. International
Journal of Systems Science, Vol. 19, 1988, pp. 2163-2168.
[6] Gersch, W. and Luo, S. Discrete time series synthesis of randomly
excited structural system response. Journal of the Acoustical Society
of America, Vol. 51, 1972, pp. 402-408.
[7] Gersch, W. and Liu, R.S-Z. Time series methods for the synthesis
of random vibration systems. ASME Journal of Applied Mechanics,
Vol. 98, 1976, pp. 159-165.
[8] Gersch, W. and Yonemoto, J. Synthesis of multivariate random vi¬
bration systems: a two stage least squares ARMA model approach.
Journal of Sound and Vibration, Vol. 52, 1977, pp. 553-565.
[9] Durbin, J. The fitting of time series models. Reviews of the Interna¬
tional Institute of Statistics, Vol. 28, 1960, pp. 233-244.
[10] Mayne, D.Q. and Firoozan, F. Linear identification of ARMA pro¬
cesses. Automatica, Vol. 18, 1982, pp. 461-466.
[11] Georgiou, T.T. Realization of power spectra from partial covariance
sequences. IEEE Transactions on Acoustics, Speech, and Signal Pro¬
cessing, Vol. 35, 1987, pp. 438-449.
[12] Lee, J.E. and Fassois, S.D. On the problem of stochastic experi¬
mental modal analysis based on multiple-excitation multiple-response
data. Part I: dispersion analysis of continuous-time structural sys¬
tems. Journal of Sound and Vibration, Vol. 161, 1993, pp. 33-56.
1547
1548
FIRST PASSAGE TIME OF MULTI-DEGREES OF FREEDOM
NON-LINEAR SYSTEMS UNDER NARROW-BAND
NON-STATIONARY RANDOM EXCITATIONS
C.W.S. Tot and Z. Ghent
Department of Mechanical Engineering
University of Nebraska
255 Walter Scott Engineering Center
Lincoln, Nebraska 68588-0656
U.S.A.
E-mail: cwsto@unlinfo.unl.edu
Abstract
The study of motion of aerospace systems and buildings, that house
sophisticated and expensive electronic equipment under intensive transient
disturbances, has in recent years become an important issue in the design and
analysis process. Central to the study is the problem of predicting motion and
first passage time of the system under such disturbances.
Owing to the intensity and nature of the excitation, and the non¬
linearity of the deformation of the complex system, techniques available for
the response predicting is very limited. To provide a more realistic and
accurate prediction of response and assessment of the first passage probability,
in this paper the extended stochastic central difference method is proposed.
The recursive response statistics and first passage time of a multi-degrees of
freedom non-linear system under excitations treated as narrow band non¬
stationary random excitations are considered. Results of a two degrees of
freedom non-linear system indicate that the technique proposed is very
efficient and accurate compared with the Monte Carlo simulation data.
Professor and corresponding author
Research Assistant, Department of Mechanical Engineering, The University of
Western Ontario, London, Ontario, Canada N6A 5B9
1549
1. INTRODUCTION
The study of motion of aerospace systems and buildings, that house
sophisticated and expensive electronic equipment under intensive transient
disturbances, has in recent years become an important issue in the design and
analysis process. Central to the study is the problem of predicting motion and
first passage time of the system under such disturbances.
Owing to the intensity and nature of the excitation, modulated wide
band random excitation process is a highly idealization and consequently, even
for a two degrees of freedom (dof) linear system it can lead to large
differences between the responses of the system under such an excitation
process and those using a more realistic representation of narrow band non¬
stationary random disturbance [1]. For a single dof nonlinear system, the first
passage time based on wide band random excitation is very significantly
different from that using the narrow-band random excitation process [2]. A
survey of the literature seems to suggest that no analytical or computational
technique, bar the Monte Carlo simulation (MCS) method, is available for the
prediction of motion and first passage time of multi-degrees of freedom
(mdof) nonlinear systems under narrow band non-stationary random excitation.
When the number of dof is large and the excitation is a narrow band non-
stationary process, even the application of the MCS can be difficult if not
impossible. The difficulty lies in the choice of centre frequency, bandwidth
and amplitude of the response from the filter in order to model the narrow
band process that is being approximated and applied to the non-linear system.
To provide a more realistic and accurate prediction of response and assessment
of the first passage probability, in this paper the extended stochastic central
difference (SCD) method is proposed. The recursive response statistics and
first passage time of the mdof non-linnear system under excitations treated as
narrow band non-stationary random excitations are considered.
The organization of this paper is as follows; a brief introduction to the
extended SCD method is given in the next section which is followed by a
consideration of the two dof system under narrow band non-stationary random
excitation in Section 3. The versatile statistical linearization (SL) or equivalent
linearization (EL) technique is applied at every time step to linearize the two
dof non-linear system. The scheme adopted'liere is scheme TV in reference [3].
Section 4 is concerned with the modified adaptive time scheme (ATS) and the
first passage probabilities. Computed results are included in Section 5 while
remarks are in Section 6.
1550
2, STOCHASTIC CENTRAL DIFFERENCE METHOD FOR NARROW-BAND
RANDOM EXCITATIONS
Consider a mdof system under narrow band random excitations, which
are obtained from output of a filter perturbed by modulated Gaussian white
noise excitations. The governing equations of motion in matrix notation are
^ff + ^ff + ^ff = > ( I a,b)
where Mf, Cf and Kf are the mass, damping and stiffness matrices of the filter
while My, Cy and Ky are the mass, damping and stiffness matrices of the
system respectively; f is the stochastic displacement vector of the filter, and
Y is the stochastic displacement vector of the system; e(t) is a vector of time-
dependent deterministic modulating functions; the over-dot and double over
dot are the first and second derivatives, respectively; w(t) is a zero mean
Gaussian white noise process.
Recursive expressions for the mdof filter and mdof system described
by equation (1) can be derived [1]. For example, the recursive expression for
system responses can be shown as
(A0‘‘N„K/s)< + N^yDyis)Nl * N,yDj(s)Nl ^2)
+ (At)'Nj^G(s)W,; + (AtfN,yG(s)'^Nl
* iAtfN,yH(.s)Nl *
where R,(s) = Ry(s) = <Y,Y7>; D (s) = <Y,Y,.,'"> = R,(s-1) + N,,
D/Cs-l) + (At)' Ni, G(s-l)'', G(s) = <Y,f7>, H(s) = <Y„,f7>, f, = f(s) is f at
time step t„ At=t„,-ts. Y. = Y(s) is Y at time step t„ the angular brackets
denote ensemble average of the enclosing quantity while the superscript T
desingates the transpose of the matrix;
N,y -IMy* (Atm Cy]-^ , ' N^y [ 2M^ " (AO^iT^] ,
N^y-N,yl{Atl2)Cy-My], -
G(s) - (AtfN^yD^(s) * N^yH(s)
+ N,yH{s-l) * N,y G(s-2)
H(s) - GCs-DW,^* (.AtfN^yRp-2)N,^f
* N^y G{s-2) Ny * N,yH(s-'i) K-
1551
Equation (2) is applicable to systems under multiple narrow band random
excitations generated by the filter matrix equation.
In order to provide the required response statistics for the first passage
probability computation, the covariance of displacement and velocity
responses, and the covariance of velocity response are considered in the
following. Starting from the central difference method, the velocity vector at
the current time step is given by
f(s) . ^[y(y+l)-7(s-l)]
2Ar
which upon post multiplying the transpose of displacement vector Y(s) gives
the covariance matrix of velocity and displacement as
( r(^) r '■(s) ) - ^ [ D(s* 1) - D ^(s) ] , (3)
since D'''y(s) = = <Y5.,YJ>. Similarly, the covariance matrix of
velocity responses becomes
f 5(s+l) + fe-l) - (AtfN H^(s)
- N^,D^(s) - N,^R^(s-D (4)
The foregoing is the so-called extended SCD method. When the excitations are
wide band stationary or non-stationary random processes the above equations
reduce to those of the SCD method. Equations (2), (3) and (4) can be applied
to the response analysis of mdof systems under narrow band non-stationary
random excitations. The corresponding first passage probability can be
evaluated as in reference [4]. For non-linear systems whose nonlinearities are
explicitly defined, however, the following technique is employed in
conjunction with the modified ATS before the computation of first passage
probabilities can be performed.
3. NON-LINEAR SYSTEMS AND STATISTICAL LINEARIZATION
Consider the two dof system shown in Figure 1. The system is
subjected to a narrow band non-stationary random excitation at its base and
the restoring force of the spring connecting the two masses Mj and M2 has
quadratic and cubic nonlinear terms associated with, respectively, parameters
1552
T]’ and e’. This two dof system under a modulated white noise excitation was
considered by Kimura and Sakata [5], and Liu and To [3, 6]. Practical
examples that can be modeled by this system are; (a) soil-structure coupled
system, (b) a primary building structure with secondary system representing
installed equipment under an earthquake excitation, and (c) an offshore oil
platform with a secondary equipment such as the drill string tower under
strong wave impact.
If one introduces relative displacements or inter-story drift in the
language of earthquake engineering such that = yi - yo and Y2 = y2 - yi,
where yg, yi and are, respectively, the absolute displacements with respect
to the base, the matrix equation of motion can be expressed as [3, 5, 6]
1 o'
jM
> +
^—4
0^
2C,W ■
2(1 + p)C2
I (xtiy/ -
r-'
« +
ri
i'. ,
-W^ 1+li
In’l
(5)
or written in a more compact form
M Y ^ C Y ^ K^Y ^ g{Y) = r(T)
(6)
in which
1 o'
\2i;,w
\ , c = \
0 1,
1 [-2C,W
2(1+vi)C2
-JV^
-P
Up
g(X) -
(h-p)Ti72^ + a +
Y =
r(T) =
e(T)v/7|
1553
where the over-dot and double over-dot denote, respectively, the first and
second derivatives with respect to T. In equation (6) the non-stationary random
excitation is represented as a product of a deterministic amplitude modulating
function e(T) and a zero mean narrow band random process which is related
to the response from the filter. The symbol I is assumed to be constant. Note
that the following definitions have been applied in equations (5) and (6)
(0^ - , <4 - , 2CiQi - Cj/Mj ,
- Cj/Mj . Ti - .
e - , V- ' ,
such that equations (5) and (6) are dimensionless. Therefore, the symbol T in
this section should not be confused with the T used in the time co-ordinate
transformation (TCT) in reference [6].
The discretization of equation (6) in the x domain leads to
M f, + C f, + Jfo r, + g(y,) - r, (8)
where the subscript s is a positive integer denoting the time step such that
Ax = X3+1 -Xj and Xq = 0.
Since this is a non-linear system, the mean of its response will in
general be non-zero even though the excitation is of zero mean. It is generally
agreed that the assumption of Gaussian response in the linearization technique
of Caughey [7] is approximately satisfied for weakly nonlinear systems. In the
present investigation the highly nonlinear systems are approximated as a series
of weakly nonlinear or linear systems and therefore one can assume that the
response at every time step is Gaussian. Experience [3,6] with nonlinear
systems under wideband random excitations shows that such an assumption is
acceptable as very accurate results were obtained. In spirit similar to
references [3,6], equation (8) can thus be represented by the following
linearized equation
1554
where K^q is the equivalent stiffness matrix which is time dependent. It is
determined by the following equation
BgfJis))
ij =1,2.
Upon substituting the non-linear term g(Y) into equation (10) and carrying out
the operation one has
0 2\ir\<Y^is)> - 3p.t<Y2is)>
K(s) - K. + , • ^ ^
0 2(l-p)Ti<y2(5)> + 3{U\i)B<Y^is)>
With equation (11), one can then apply the SCD technique to equation (9) and
perform the operation to obtain the recursive expression which is identical to
that described in Section 2 above. The modified ATS to be introduced in the
next section has to be employed with the SCD method to update the K^q at
every time step.
As the system is non-linear and therefore ensemble averages of
responses are not zero in general, however, the following recursive ensemble
average of response vector clearly indicate that it is zero if the system starts
from rest
m(5+l) = N2y(s)m(s) + N^ym(s-l) . (12)
where m(s+l) is the ensemble average of system response vector at the next
time step.
To circumvent this problem, the g(Y) term of equation (6) is moved
to the right hand side (RHS) such that
M ns) * C y(s) + r(s) - r(s) - g(r(i)) . (13)
As will be seen in the following this equation is applied only for the initial
states of the response predictions. Substituting the central difference
approximation of velocity and acceleration Jerms
y(s) - — [ y(s+i) - y(s-i) ] ,
2 At
y(i+i) - 2 ns) * i'(s-i)] .
(14)
(15)
into equation (13) one has
y(i+l) - N^^Y(s) + JVj, 7(5-1) +
- (AT)2Wi^g(y(5)) ,
where
Nj, = [ 2M - iAx)% ] .
(16)
Taking the ensemble average of equation (15) one obtains
mis^l) - mis) + m(5-l) - {Ax)^ N^y<giYis))> ,
<g(Yis))>
^nl<y2(5)^> -
n e [ 3 <y,(5)><yj(s)^> - 2 <y2(5)>^ ]
(i-ti)Ti<yj(s)"> f
(i+n)e[3<y2(5)><y2(5)2>-2<yj(5)>^]
(17)
Substituting equation (17) into equations (2) and (13) one can show
that
R (2) . [Ax)^N,^RJiDN^/ , m(2) - [0 0]^,
(l6)
m(3) - -(AT)S<l'2W^>iV,,[p !-(»]’■•
As soon as the above non-zero ensemble averages are found, one can return
to equation (12). For s > 3, equation (12) can be applied to obtain the non¬
zero ensemble averages. Of course, the recursive mean squares are given by
equation (2). Up to this stage, the SCO method with modified ATS technique
can be applied to the system to compute its responses to random excitations.
4. MODIFIED ADAPTIVE TIME SCHEME AND FIRST PASSAGE PROBABILITIES
As described by Liu and To [6], the time step size in a non-linear
system varies with the variance of displacement response, the time step size
has to be updated accordingly. This strategy is known as the adaptive time
scheme (ATS). In the present analysis the terms that have to be considered for
1556
the time step updating are Ry(s-l), G(s-2), H(s-2), Rf(s-2), Niy, N2y and N3y.
This has become more critical in the extended SCD method because the error
would come from not only the system but also the responses of the filter. The
interpolation scheme is employed to compute the time step size when it is
warranted to do so for the nonlinear system whose natural frequency at every
time step is different.
With the variances of displacement and velocity at every time step
computed by the extended SCD method, the first passage problem can now be
considered. Approximate first passage probabilities based on the modified
mean rate of various crossings proposed earlier by the first author [4] are
computed. In the latter the trapezoidal rule was employed to evaluate the first
passage probabilities and uniform time step was assumed. For nonlinear
system employing the SCD or extended SCD method and modified ATS, the
time steps are different. Therefore, the interpolation scheme is applied in the
present investigation such that the trapezoidal rule can be adopted for the
computation of first passage probabilities.
5. NUMERICAL RESULTS
Numerical examples are presented in the following. Parameters of the
system in Section 3 are:
IF = p = 1.0 , Cl = ^2 =
such that the two dimensionless natural frequencies of the corresponding linear
system, that is when T\ and s are both equal to zero, are cOji = 0.618 and (£>,2
= 1.618. The amplitude modulating function e(T) chosen is
e(T) - - e-o") . ^20)
Numerical results presented in this section include two examples. The
first example is the one with the centre frequency of the one dof cOf = 1.0 and
the second is with cOf = 1.618. Each example has a two dof system described
in Section 3 and a single dof filter. The latter equals to one of the two linear
dimensionless natural frequencies of the, non-linear system. The other is
between the two linear dimensionless natur^ frequencies. Damping ratio of the
filter (band width) in the two examples are = 0.01, 0.1 and 1.0. The narrow
band random excitation to the system is the response taken directly from the
filter which is perturbed by a non-stationary zero mean Gaussian white noise.
The spectral density of the Gaussian white noise is Sq = 0.00012. The narrow
band random excitation to the system is at its base. MCS is also carried out
in order to verify the results by the extended SCD method. Other pertinent
system parameters are t] = -1.0 and e = 1.5.
1557
It was observed that the results of the case where cOf = 1.0 and =
0.01 became unstable for both the extended SCD and MCS methods. An
attempt was also made to compute responses of the system excited by a
narrow band process with cOf = 0.618. In this case, instability occurs when
equals to 0.01 and 0.1. Note that the spectral density of the white noise
process is one order of magnitude smaller than what was used in reference [6].
This is because when Sq = 0.0012 was used, the response of the system
becomes unstable. Representative response statistics for Sq = 0.0012 are
present in Figures 2 and 3 in which the covariances of displacement and
velocity responses were not included for clarity since they are very close to
the variance <Y2>- From the figures, one observes that the results by the SCD
method has an excellent agreement with those using the MCS. The
computational time using a Silicon Graphics Inc. engineering workstation with
64 megabyte random access memory and 60 mega Hz single central processor
the extended SCD method is approximately 1 5 seconds while that for the MCS
is about 57 minutes. Consequently, one can conclude that the extended SCD
method is very efficient and accurate compared with the MCS data.
For the first passage probabilities, two sets of results were studied.
They are: (i) results concerned with narrow band non-stationary random
excitations, and (ii) comparison of narrow band non-stationary random
responses to wide band non-stationary random responses. It was observed that
in term of first passage probability the difference between narrow band non-
stationary and narrow band stationary random excitations is insignificant and
therefore these results are not presented here. In set (i), results obtained by the
MCS technique are included for direct comparison. Some typical results of
first passage probability Ld based on the two stage process with allowance for
the actual duration of clumps at low threshold levels are presented in Figures
4 and 5. The results for set (ii) are plotted in Figures 6 and 7. In these figures
NB designates narrow band, and WB wide band. With reference to the results
in Figure 7, one can conclude that the first passage probability of the model
with narrow band random excitation is very much different from that with
wide band random excitation.
Before leaving this section it may be appropriate to note that although
the filter and system parameters for sets (i) and (ii) are identical the
corresponding plots in Figures 5 and 7 are different. For the results in set (ii)
and plots in Figures 6 and 7 the input ^ the system is a product of an
envelope function and a stationary narrowband random process. However, for
the results in set (i) and plots in Figures 4 and 5 the input to the system is the
output of the filter whose input is a product of an envelope function and a
zero mean stationary white noise process. In other words, in the MCS one is
unable to have an absolute control over the exact narrowband nonstationary
input to the system. The extended SCD method, however, has no such a
restriction and therefore has an added advantage over the MCS in controlling
the input to the system.
1558
6. REMARKS
In this paper the extended stochastic central difference method is
introduced. The recursive response statistics and first passage time of a multi¬
degrees of freedom non-linear system under excitations treated as narrow band
non-stationary random excitations are considered. Results of a two degrees of
freedom non-linear system indicate that the technique proposed is very
efficient and accurate compared with the Monte Carlo simulation data.
One also observes that the first passage probability of the two dof
model under narrow band random excitation is very much different from that
with wide band random excitation. This suggests that correct representation of
the excitation process as a wide band or narrow band random process is
extremely important if more reliable conclusions are to be drawn and used in
the design process.
ACKNOWLEDGEMENT
The investigation reported above was supported in the form of a
research grant by the Natural Sciences and Engineering Research Council of
Canada awarded to the first author.
REFERENCES
1. To, C.W.S. and Chen, Z., Response analysis of system under
narrow band stationary and nonstationary random excitations. In Proceedings
of A.S.M.E. Pressure Vessels and Piping Conference, Montreal, Canada, July
21-26, 1996, PVP-Vol. 331, pp. 107-114.
2. To, C.W.S. and Chen, Z., First passage time of nonlinear ship
rolling in nonstationary random Seas. In Proceedings ofA.S.C.E. 7th Specialty
Conference on Probabilistic Mechanics and Structural Reliability, Worcester,
Massachusetts, U.S.A., August 7-9, 1996, pp. 250-253.
3. To, C.W.S. and Liu, M.L., Recursive expressions for time
dependent means and mean square responses of a multi-degree-of-freedom
nonlinear system. Comput. Struct., 1993, 4^ (No. 6), 993-1000.
4. To, C.W.S., Distribution of the first passage time of mast antenna
structures to nonstationary random excitation. /. Sound and Vibration, 1986,
108, 11-23.
5. Kimura, K. and Sakata, M., Nonstationary response analysis of a
nonsymmetric nonlinear multi-degree-of-freedom system to nonwhite random
excitation. JSME Int. J., 1988, 31(4), 690-697.
1559
6. Liu, M.L. and To, C.W.S., Adaptive time schemes for responses of
nonlinear multi-degree-of-freedom systems under random excitations. Comput.
Struct., 1994, 52(No. 3), 563-571.
7. Caughey, T.K., Equivalent linearization techniques. J. Acoust. Soc.
Am., 1963, 35, 1706-1711.
V.
Figure 1. The two dof non-symmetric non-linear system
1560
1564
RANDOM RESPONSE OF DUFFING OSCILLATOR
EXCITED BY QUADRATIC POLYNOMIAL OF
FILTERED GAUSSIAN NOISE
C. Floris and M.C. Sandrelli
Department of Structural Engineering, Politecnico di Milano, Piazza
Leonardo da Vinci 32, 1-20133 Milano, Italy
ABSTRACT
The random response of a Duffing oscillator excited by a quadratic
polynomial of filtered Gaussian process is considered. The problem is
approached by the use of Ito's stochastic differential calculus. Being
the system nonlinear, the moment equations (ME) constitute an infinite
hierarchy, to close that a procedure different from the classical cumu-
lant-neglect closure method is used. This procedure operates in two
phases, in the former of which the system is linearized so that the ME
are solved exactly. In the second phase the actual system is considered:
the equations for the moments of m-th order contain moments of order
m+2. To solve them an iterative scheme is used, in the first step of
which the higher order moments are considered as known quantities
and take the values that have been estimated on the linearized system.
In the applications the new procedure is shown to converge towards
the solution obtained by simulation requiring a charge for calculation
considerably lesser than that of cumulant-neglect closure method. Care
is taken to detect possible multiple solutions when the excitation is
narrow-banded.
1. INTRODUCTION
The study of nonlinear dynamical systems under random excitation
of polynomial form is of considerable interest in the theory of random
vibration as well as in engineering applications. In fact, important ran¬
dom dynamical agencies can be properly represented as polynomial
forms of filtered Gaussian processes. This is the case of gusty wind [1]
and random sea waves [2-4]. In this way, both the system and the exci¬
tation are nonlinear, which increases the difficulties in characterizing
the response statistically. System response is probably well far from
Gaussianity, since the non-normality is caused by both the non-
1565
normality of the input (memoryless transformation of normal process)
and non-linearity of the response process itself.
Nonlinear random vibration problems can be approached by sever¬
al ways, such as equivalent linearization, perturbation, quasi-harmonic
method, multiple time scales, and Ito's stochastic differential calculus.
In this paper the last approach is followed. It is possible since the exci¬
tation is idealized as the output of a linear filter excited by a Gaussian
white noise. Ito's stochastic calculus provides a straightforward proce¬
dure for deriving the differential equations in terms of response mom¬
ents or in terms of the probability density function [5-9]. Nevertheless,
in presence of nonlinearities in both the excitation and the system man-
y difficulties arise. The so-called Fokker- Planck-Kolmogorov (FPK) e-
quation, which gives the probability density function of response, can
be solved analitically in very few cases, especially when the transient
response is considered (e.g. see [10]). If the moment equation (ME) ap¬
proach is used, the equations of moments of order k involve higher and
lower order moments than k, that is the ME constitute an infinite hier¬
archy. In order to solve this problem closure schemes have been pro¬
posed [9,11,12]. Among the closure schemes the cumulant-neglect clo¬
sure method is certainly the most popular. However, although this
method is generally efficacious, it has some shortcomings. The
moments of order higher than k that appear in the equations for the
moments of order k are expressed as a function of the lower order
moments setting the corresponding cumulants equal to zero. Since
these functions are nonlinear, the ME too become nonlinear. In such a
way computational difficulties arise. Moreover, a nonlinear system
may have more acceptable solutions (it is recalled that a solution is
acceptable when ail even moments are positive and Cauchy-Schwartz
inequality is satisfied). These multiple solutions must be considered
with care, possibly determining whether they are stable or not. The
significance of multiple solutions in terms of statistical moments of the
response of a nonlinear dynamical system is still an open question [IS¬
IS]. However, in some cases such as a Duffing oscillator multivalued
response moments are expected in the presence of a narrow-band
excitation [17]. These different values of variance correspond to differ¬
ent local states around which the system oscillates with abrupt jumps
between the different states. Furthermore, the steady-state probability
density function (PDF) of response is bimodal or multimodal, even if it
is unique, since FPK equation has one solution only. In other cases that
have been studied by the writers and Prof. Di Paola [18] the simulated
response does not show any jump and the significance of multivalued
response is not clear.
1566
In this paper a new closure scheme is adopted. The procedure oper¬
ates in two phases, in the former of which advantage is taken of the
fact that the ME of linear systems excited by polynomial forms of
Gaussian processes [2, 19] or delta-correlated processes [20,21] are ex¬
actly solvable. Hence, the nonlinear system is replaced by a linear one,
whose parameters are chosen in such a way to minimize the error
between the original and the linearized system in some statistical sense
[22,23]. The statistical moments, which are exact for the latter, are cal¬
culated to an order larger than that one chosen for the original system.
The reason for this choice will be explained in the next section. In the
second phase, the ME of the original system are considered. Now, the
procedure operates iteratively: in the first step the higher order mom¬
ents that appear in the equations for the moments of k-th order are con¬
sidered as known quantities taking the values that have been eval¬
uated for the linearized system. In the generic i-th step these moments
are set equal to the values obtained in the step z-1. The iterations are
carried on till the moments do not change appreciably. This procedure
is applied to a Duffing oscillator excited by a quadratic polynonial of a
filtered Gaussian process. The damping and the nominal vibration pe¬
riod of the oscillator are kept constant, while the bandwidth of the fil¬
ter is varied in order to ascertain the effects of this quantity on the re¬
sponse. The results of the proposed approach are compared with those
of Monte Carlo simulation. When the bandwidth of the filter is narrow,
eventual multivalued responses are searched using either the
stochastic differential calculus or Monte Carlo simulation.
2. PRELIMINARY CONCEPTS
In the study of nonlinear vibrations the Duffing equation plays a
fundamental role. In normalized form it reads as
x(t) + f>Qx(t) + (£>Q^x(t) + sx(t)^'^ = F(t) (1)
in which is the coefficient of viscous damping, cOq is the pulsation of
the corresponding linear oscillator (e = 0), and the term £x( t f repre¬
sents the nonlinear restoring force, acting as a hard or soft spring for
positive or negative values of 8, respectively. In random vibration
studies F(t) is generally assumed to be a Gaussian process, since this
type of stochastic process allows the use of analytical methods. When
Fit) is a Gaussian white noise, the reduced Fokker-Planck-Kolmogorov
equation [10] associated with (1) admits an analytical solution [24] as
follows
1567
p(x,x) = Cexp
(2)
in which p(x,x) is the steady-state joint probability density function
(PDF) of x,x} C is a normalization constant, and zv,, is the strength of
white noise, which is expressed as F(t) = ^KWQW( t ), being W(t) a unit
strength Gaussian white noise.
However, not all excitations are so broad-banded that they can be
idealized as a white noise. Very narrow-banded excitations can be
considered as monochromatic, say
F(t) = A(t)cos(0 ft + B(t)sin(ii ft (3)
in which A(t) and B(t) are two slowly varying Gaussian processes.
Otherwise, a frequent idealization of F( t) is thinking it as the output of
a second order linear filter, say
F(t)~\jf(t) yf(t) + ^ fyf(t) + (iijyf(t) = ^KZUQW(t) (4)
where W(t) is a unit strength Gaussian white noise. The steady-state
FPK equation associated with the augmented system of (1) and (4) has
not an analytical solution, since this system is not amenable to the
classes of detailed balance or of generalized stationary potential [10].
However, the primary excitation in (4) is a Gaussian white noise, so
that the problem can be suitably framed in the context of Ito's
stochastic differential calculus [5-9].
Likewise to the deterministic response of Duffing oscillator sub¬
jected to a sinusoidal excitation, in which there appear phenomena
such as multiple valued, subharmonic and superharmonic, responses
and the associated jumps in response levels, for sufficiently narrow
bandwidths of the excitation the random response can exhibit multiple
values of the statistical moments, particularly of the variance. This fact
was revealed by the analyses performed by some researchers that used
different approaches. Some used equivalent linearization technique
[13,14,25-27]. Lennox and Kuak [28] faced the problem through a
combination of deterministic and stochastic averaging. Other authors
preferred the methods of multiple time scaling, quasi-harmonic
analysis and harmonic balance [29-31]. More recently, Ito stochastic
calculus with a non-Gaussian closure was used in Ref. 16. However,
the significance of the multiple solutions generated by equivalent
linearization and Ito's calculus is still an open question. The
discussions about this subject in Refs. 15, 16 are remarkable.
This paper concerns the response of Duffing oscillator (1) to an ex¬
citation of the form
F(t) = aQ+aiyf(t) + a2yf(tf (5)
1568
being yjit) the output of a filter such as (4). It is recalled that (5) can
represent wind excitation by properly choosing the coefficients a^-a^. In
a literature search made by the writers no theoretical and applied stud¬
ies have been encountered regarding the response of Duffing oscillator
to excitations of the form (5) or of the more general form
Fit) = flo + a^y fit) ^ a^y fit)^ ^ . (6)
Clearly, the problem is rendered more difficult by the twofold non¬
linearity in both the structure and the excitation. If Ito's calculus is
applied, the ME form an infinite hierarchy with many higher order
terms arising from the term in (1) and the term ajyj in (5). In this
way, a large number of cumulant-moment relationships must be set
equal to zero to close the hierarchy, making the ME highly nonlinear.
Furthermore, these last are more than two hundred, if the statistical
moments of fourth order are to be computed. Even if the steady-state
solution is of concern, and, hence, the ME become algebraic, it is
cumbersome to find a solution and more physically realizable
solutions may exist. In order to overcome these difficulties, advantage
is taken from the fact that the statistical moments of a linear system
excited by a polynomial form of a filtered Gaussian process are
calculated exactly [2, 19-21]. Therefore, in the first step of the
procedure that is here proposed the system (1) is linearized. The
statistical moments that are computed in this way constitute a first
estimate of the moments of the actual system and are the basis of the
algorithm with which the hierarchy of ME is closed.
3. ITERATIVE PROCEDURE
3.1 Linearized system
In the first phase of the iterarive procedure Eq. (1) is substituted by
xit)+pQxit) + (i)^xit) = Fit) (7)
in which Fit) is given by (5), and is the linearization parameter.
Making the mean square error between (7) and (1) minimum [22, 23] it
is got
cOg =o)o +ecoo •e[x^]/e[x^] (8)
in which £[•] denotes ensemble average. Since the linearization is
confined to the structure, and the excitation holds the nonlinear form
(5), xit) is not a Gaussian process so that the relationship E[x^] - 3(£[x‘]-
E[x]f is not valid. Being E[x‘‘] unknown a priori, the equivalent
linearization is applied iteratively starting from a tentative value of the
1569
fourth moment of x. The statistical moments are computed to the
fourth order, and a new value of co^" is calculated by using (8). The
iterations are terminated when at step n |cOg „ - co^ <8^, being 5e of
the order 10*^ It is recalled that the non-Gaussianness of the response
makes it very difficult to derive an analytical relationship for response
variance as it is done in Refs. 25,26. Thus, the search of eventual
multiple values of variance is made numerically.
In order to compute the statistical moments of the response, Ito's
differential rule is applied [6-9]. The state variables x^-x, X2-x ,x^~
y^, and x^-y^ are introduced and Eqs. (4,7) with the excitation (5) are
written in incremental form, say
dxi =X2'dt (9a)
dx2 = -^qX2 - dt -(£)^x-^ ■ dt + Uq ■dt + a-^x^ • dt + ^2^3 • dt (9b)
dx^ -x^-dt (9c)
dx4 = -P fX^ • dt - C0yX3 • dt + ■ dB (9d)
in which dB is the increment of a unit Wiener process (Brownian mo¬
tion) that is related to white noise by the formal relationship dB/dt =
W(t).
The stationary first order moment (statistical average) of x is
E[i] = E[j:i ] = {uq + ]} (10)
E[x] depends on the mean square value E[x3j = £[y?j, that is known,
being y^ a Gaussian process.
The equations for the stationary second order moments read as
2E[xiX2] = 0 (11a)
-2p oE[^2 ]~ e E[^i^2 ] + 2aQE[x2]+ 2aiE[x2X^ ] + 2fl2E[^2^3 ] = 0
(11c)
In Eqs. (11) there are moments of order lower and higher than the
second. The former, E[xi] and £[^2]/ are known. Viceversa, the
moments of the types E[ziZj^] (2 = 1,2; k = 3,4), and j (2 = 1,2;
r+s = 2) are unknown. However, Eqs. 11 do not constitute an infinite
hierarchy even if there are third order moments. The technique pro¬
posed by Muscolino [20] is used to solve the problem. It requires the
solution of two linear and uncoupled sets of equations that are written
1570
for the moments E[ziZj^], and respectively. Analogously, the
evaluation of the moments of third and fourth order of xi,X2 requires
the solution of five and six linear uncoupled sets of equations,
respectively. Once convergence is obtained for co J , the moments of the
linearized system are computed to the sixth order. In this way, a first e-
stimate of the response moments is obtained.
3.2 Nonlinear system
Now we return to the actual system driven by Eq. (1). Eqs. (1,4) are
written in incremental form: Eq. (11b) is replaced by
clx2 = -p 0X2 ■ dt - (£)lxi • dt - ECO gxf ■ dt + Uq ■ dt + ■ X2 • dt + a2 ■ xl • dt
(12)
The ME of the actual system are written applying Ito's differential rule
again. They constitute an infinite hierarchy really, since in the
equations for the moments of k-th order there are moments of order
k+2 because of the term in (12). In this paper a new technique is
used to close the hierarchy profiting by the solution that has been
obtained for the linearized sys tern. This technique operates iteratively.
Let us examine some ME of the actual system. The statistical aver¬
age is given by
E[j:i] = E[x] = cOo^{ao +fl2E[x|]}-eE[xq (13)
E[x] depends on the third order moment j, which in the first step
is assumed to be known and take the value that has been computed for
the linearized system. In the other iterations it takes the value of the
previous one.
The equation for a moment of order m can be written as
ii-pq =^PP-p^l,q+l-^?>0\^pq-<^(^l\^p+l,q-l ' (i^(^0P-p,q+2 +Wp.q-l +
in which p + q = m and yipcj- structure is exposed to a
stationary excitation for a sufficiently long time, it reaches the sta-
tionarity and the l.h.s. of (14) vanishes. In Eq. (14) there are different
types of moments: The fifth addendum has an order lower than m and
has been already evaluated, while the first three addenda are of the m-
th order, and as many equations as these moments can be written.
1571
However, the fourth term and the last two are additional unknowns.
The cross-moments among x^, and the variables of the filter are eval-
luated writing the corresponding equations down, as for the linearized
system, and the hierarchy is overcome. The fourth addendum is
an effective hierarchical term. In the first step of the iterative proce¬
dure it takes the value that has been computed for the linearized
system, while in the generic i-th iteration it takes the value of the
iteration z-1. In this way, the equations for the moments of different
order and type are considered separately with an evident computa¬
tional advantage. Furthermore, the ME remain linear, while in the
cumulant-neglect closure method [11, 12] the cumulant-moment rela¬
tionships introduce non-linearities of degree as higher as the level on
which the closure is made. In the case under examination with four
state variables, computing the moments to the fourth order, there
would arise more than two hundred equations, and this large nonlin¬
ear set of equations would be hardly solvable with a personal comput¬
er. Viceversa, using the proposed approach the response moments are
computed by solving linear systems with a small number of equations.
Since the highest order moment appearing in (14) has the order 7n+2
and the moments of the actual system are computed to the fourth
order, the moments of the linearized system are evaluated to the sixth
order.
3.3 Search for multiple solutions
Since the writers did not find any study on Duffing oscillator ex¬
cited by a polynomial form of filtered Gaussian process, it was not
possible either to affirm or to exclude the possibility of a multivalued
variance in the presence of a narrow-band excitation. Hence, there is
need of ascertaining whether the response moments under a given
excitation are unique or not. The polynomial form (5) of the excitation
does not leave room to a simple expression for the variance as in the
case of a filtered Gaussian excitation [13,14,25-27]. Moreover, the
bandwidth of the excitation cannot appear explicitly in this case. Thus,
an iterative-numerical procedure is used for searching multivalued
responses.
The search is made with reference to the linearized system, since it
is simpler to be done. From Eqs. (11) we find for the mean square
values of x^ and x,, respectively
] = CO 7^ |e[x| ] + aoE[x-^]+a^E[xiX3 ] + a2E^x-ixl ]} (15a)
1572
%!]= Po^{^^iE[^2^3] + «2%24]} (15b)
Eq. (8) giving the linearization parameter is written as
co^ =cOo|l. + £fcE[^i ]j- The procedure must accomodate for the possi-
<2^
bility of a multivalued variance. After obtaining convergence for cOg
and computing the statistical moments, the ratio k = £[Xj'']/E[x/]^ is
calculated and substituted in the expression of co g , which at its turn is
inserted in Eq. (15a). In Eqs. (15) there are the moments £[xj, E[xx^l
and E[xpc^] (z=l,2) that obviously depend on cOgi As a first approx¬
imation these moments are assumed to take the values obtained at the
end of the iterations for finding (^. In this way (15a) becomes a quad¬
ratic equation in the unknown £[Xi“]. Obviously, the two roots must be
real, but from a theoretical point of view it carmot be excluded that
both result positive. If this were the case, the iterative procedure would
be reentered and split into two different searches. The two sets of sta¬
tistical moments that would be computed would become two different
bases for the iterative search operated on the actual system. In this way
the iterative computation of the moments of the actual system becomes
twofold, but a -priori it is not possible to affirm whether the con¬
vergence is towards two different states or towards the global state
corresponding to the unique solution of the FPK equation. The re¬
search for a multiple solution is made by simulation too, as will be
explained in next section.
4. APPLICATIONS
The procedures outlined in previous sections have been applied to
a Duffing oscillator characterized by the following values of the
parameters in Eq. (1): po = 0.6283, which corresponds to a ratio of
critical damping of a 5 %; coo = 2tc; £ = 0.5. The coefficients of Eq. (5) are:
flg = 10, aj = = 1. The filter (4) is characterized by co^ = 271, and = 5.
As regards p^^ it takes the values 0.1257, 0.3770, 1.2567, which correspond
to relative dampings of 0.01, 0.03, and 0.10, respectively. The bandwidth
parameters associated with these two values are 0.1131, 0.1949, and
0.3386, respectively. The bandwidth parameter is defined as
= being Xj the spectral moment [32]. In the case
of Eq. (4) the formula q = (1.16^°-^^ -0.21)^-^ gives an accurate estima¬
tion [33]. It is remarked that the set of parameters co^ = 27C, p^ = 0.1257,
1573
Table 1; results for Duffing oscillator^ 0.1257
Elx]
■SSH
(1)
0.258459
0.105127
0.050338
0.028410
(2)
0.273143
0.073133
0.024813
0.008744
(3)
0.281231
0.080166
0.023150
0.006769
Stochastic differential calculus: (1) linearized system, (2) actual system.
(3) Simulation
Table 2; results for Duffing oscillator,. P, = 0.3770
e[x^]
E[x^]
(1)
0.078837
0.028604
0.011598
(2)
0.253232
0.073950
0.023958
0.008564
(3)
0.257975
0.067416
0.017316
0.004505
(l)-(3) as in Table 1.
Table 3; results for Duffing oscillator^ P,= 1.2567
E[x]
E[x^-]
HBUII
(1)
0.246996
0,064276
0.017517
0.004976
(2)
0.248568
0.064627
0.017491
0.004913
(3)
0.254055
0.064595
0.016426
0.004180
(l)-(3) as in Table 1.
and Wg = 5 would cause a multivalued response with a large
probability if the filter output were directly applied to the structure.
The steady state response only is considered computing the statistical
moments to the fourth order.
The results obtained by the use of the procedure proposed in this
paper are checked against numerical simulation. It has not been
possible to perform a comparison with the standard cumulant-neglect
closure method, for both the hardware and the software available to
the writers were inadequate for solving a nonlinear algebraic system of
more than two hxmdred equations. Two types of simulations have been
performed, the former of which is a standard Monte Carlo simula¬
tion. Samples with 10,000 histories of motion starting from zero val¬
ues of both initial displacement and velocity are constructed and
analyzed using ordinary statistical tools. The motion lasts for 30 s and
Eq. (1) is solved by applying a fourth order Runge-Kutta scheme.
However, the classical simulations cannot reveal multivalued
responses, since the averages are performed on a large number of
1574
Fig. 1 - Power spectral
density of the output of
filter (4), 13^= 0.1257.
10.00 —1 y(t)
0.00 400.00 800.00 1200.00 1600.00
0.00 400.00 800.00 1200.00 1600.00
Fig. 2 - Top: typical time history of the output of filter (4). Bottom: re¬
sponse displacement x{t), (3^= 0.1257.
samples. Roberts [15] suggested another type of simulation, in which a
limited number of motion histories is simulated, but each history lasts
for a relative long time and is analyzed alone. Recently, Cusumano and
Kimble [34] proposed the so-called stochastic interrogation experimen-
method, in which, differently from Robert's method, the histories of
motion have random initial conditions. Furthermore, Cusumano and
Kimble consider large samples and trace the Poincare maps to reveal
the presence of basins of attraction. Due to hardware limitations, these
two types of simulation are performed with samples of 10 histories of
1575
1.20
0.80 -1
0.40
x(t)
0.00
-0.40 H
4-
0.00 200.00 400.00
Fig. 3 - Response displacement x(t), Py = 0.3770.
I t(s)
600.00
motion. When the initial conditions are random, the initial displace-
cement and velocity are independent normal variates with zero mean
and unit variance.
Tables 1-3 show the principal statistical moments obtained by using
the three methods, for = 0.1257, 0.3770, and 1.2567, respectively. The
statistical linearization fails to yield accurate estimates of the statistical
moments for the case p^ = 0.1257. In the other two cases the results for
the first two moments are quite acceptable, while the estimates of both
the third and fourth moment are too high. In general, the proposed
approach for closing the hierarchy of the ME constitutes a substantial
improvement over the linearization even if the third and fourth
moment of the intermediate case are greatly overestimated (by a 38.4 %
for Efx^] and by a 90 % for E[x^]). Clearly, as higher the order of a
moment is, as worse the approximation of the stochastic differential
calculus. Using a higher order of closure should improve the accuracy,
but this requires the use of more powerful computers.
The performance of the proposed approach is noticeable in the first
case, say P^ = 0,1257. This case is characterized by a very narrow-
banded output of the filter (4) (Fig. 1), even if the a simulated history
of yjit) does not resemble a sinusoid (Fig. 2, top). Performing the search
for multiple solutions, Eq. (15a) has two real roots, but one of them
only is positive. Nevertheless, examining a single motion history
lasting for 1,500 s (Fig, 2, bottom) there appears a strange phenomenon.
After exposing the system to the excitation for almost 400 s, one can
note that evenly spaced peaks rise, whose periodicity is 90-100 s about.
During these peak phases the oscillations occur around the unchanged
mean value, but their amplitude increases time after time and seems to
be unbounded. The writers think that it is improper to define this
phenomenon as a multivalued response; rather, this behavior might be
assimilated to a kind of instability. An ever increasing amplitude
means an increasing mean square value that, perhaps, might be
detected by making a non stationary analysis. Thus, the statistical
moments that have been calculated with both stochastic differential
1576
calculus and simulation, whose motion histories last for 30 s, are
correct for the stationary segment of the response, only. The same
trend can be noticed for the case = 0.3770, but it is slightly faded.
5. CONCLUSIONS
A method is presented to close the hierarchy of the ME of nonlinear
systems excited by polynomial forms of filtered Gaussian processes.
This method operates in two phases in the former of which resort is
made to statistical linearization that yields a first estimate of the
moments of the non-Gaussian response. In the second phase the ME of
the actual system are considered. They constitute an infinite hierarchy
to close which, instead of using the classical cumulant-neglect closure
method that may be cumbersome from a computational point of view,
an iterative procedure is used. This is based on the estimate of the
response moments that has been obtained for the linearized system.
The proposed method is applied to a Duffing oscillator excited by a
quadratic polynomial of a filtered Gaussian process. The bandwidth
parameter of the filter is varied from a very narrow one to a medium
one to ascertain whether the response may be multivalued as in the
case of a simple narrow-banded filtered excitation. In the case of a
narrow-banded excitation a phenomenon different from those reported
in literature occurs. After a certain time system response seems to lose
the stationarity and periodically shows peaks with ever increasing
amplitude. This phenomenon cannot be detected either by the
stochastic differential calculus, if the ME are solved for the steady
state, or by the customary simulation in which the histories of motion
have a brief duration. In order to reveal this behavior it is necessary to
study responses lasting some hundreds of seconds. However, the
response moments that are given by the proposed approach are
substantially correct for the stationary segment of the response.
ACKNOWLEDGEMENTS
The authors are indebted to Prof. Mario Di Paola of the University of
Palermo, Italy, for his continuous advice and encouragement.
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tural Dynamics, eds. Ariaratnam S. T., Schueller G. I. and Elishak-
1577
off Iv Elsevier Appl. Sc. PubL, London, 1988, 181-195.
3. Li, X.-M., Quek, S.-T., and Koh, C.-G., Stochastic response of off¬
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4. Benfratello,S., and Falsone,G., Non Gaussian approach for stochas¬
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Chaotic and Stochastic Behavior, CISM Course no. 340, Casciati F. ed.
Springer Verlag, Wien, 1993, 29-92.
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Structural Systems. Prentice Hall, Englewood Cliffs, 1993.
10. Lin, Y.K., and Cai,G.Q., Probabilistic Structural Dynamics: Advanced
Theory and Applications. Me Graw Hill, New York, 1995.
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oscillators under random parametric and external excitations. Int.
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-Gaussian filtered processes. Prob. Engrg. Mech., 1995, 10(1), 35-44.
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esses and quasi-linear systems./. Appl. Mech. ASME,1997, in press.
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amic systems subjected to white random excitation. /. Acoust. Soc.
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27. Koliopulos, P.K., and Langley, R.S., Improved stability analysis of
the response of a Duffing oscillator under filtered white noise.
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of a Duffing oscillator to narrow-band random excitation. /. Sound
Vibr., 1988, 123(3), 497-506.
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monic response: multiple time scaling of a Duffing oscillator. /.
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to random vibration. /. Engrg. Mech. Div. ASCE,1972,98(2) ,425-46.
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1579
1580
EXTREME RESPONSE ANALYSIS OF NON-LINEAR
SYSTEMS TO RANDOM VIBRATION
S. McWilliam
Department of Mechanical Engineering, University of Nottingham,
Nottingham, NG7 2RD, England.
ABSTRACT
This paper considers the numerical solution of the Fokker-Planck-Kolmogorov
(FPK) equation which governs the transition probability density function (pdf)
for a certain class of non-linear system. In order to provide a numerical solution
for the pdf which is well behaved at its tails (and thus better suited to extreme
value problems), the FPK equation is transformed to give an equation for the log-
pdf. The resulting non-linear partial differential equation is solved using a
weighted-residual approach. This technique is applied to first and second order
systems, including the Duffing oscillator. The accuracy of the technique is
assessed by comparison with analytic solutions and numerical simulation.
1. INTRODUCTION
Many different types of structure, such as offshore structures, are subjected to
random loading. In the design of these structures it is particularly important that
the statistics of the response are obtained so that the maximum stresses and
fatigue life of the structure can be predicted. For the situation when the system
is linear and the excitation is Gaussian the response statistics are well-known.
However, most practical engineering structures are non-linear to some extent and
the problem of predicting the response statistics is much more difficult.
For systems which are subjected to white-noise excitation, the response is a
Markov process and the transitional probability density function (pdf) is governed
by the Fokker-Planck-Kolmogorov (FPK) equation. For other systems the FPK
equation can be applied provided that the response is a higher order Markov
process. In both these cases the problem of predicting the response statistics
reduces to solving the FPK equation for the transitional probability density
function. The difficulties associated with finding exact solutions to this equation
have lead a number of authors to develop approximate solution procedures [1-3].
Although these techniques are limited to the analysis of low order systems, they
have been applied to a number of practical engineering problems (e.g.[3,4]).
The most commonly used numerical solution procedures are the Galerkin
technique [1], the Finite Element method [2] and the path-integral method [3].
1581
Wen [1] developed a Galerkin method for the solution of the non-stationary FPK
equation, in which the joint probability density function was expanded in terms
of Hermite polynomials. The disadvantages of this technique are that it tends to
have a low-rate of convergence for systems which are strongly non-linear and the
accuracy of the tails of the distribution is not guaranteed - yielding negative
values . In contrast, the Finite Element method [2] provides a more robust
numerical solution procedure which, in principle, has the potential to deal with
all types of non-linearity and give an accurate prediction of the tails of the
distribution. However, a large number of elements, and hence a large amount of
cpu time, are needed to give sufficient resolution over the tails of the distribution
in order for it to be used for the prediction of extremes. The path-integral method
proposed by Naess [3] has recently been shown to provide an accurate means of
estimating the response pdf, including the tails. However, this too requires a
substantial amount of cpu time, and is very much at a developmental, stage.
In contrast to the above techniques, the primary aim of the present work is to
accurately estimate the tails of the response distribution. This is achieved by
considering the log-pdf, which has the advantage of ensuring that the resulting
probability density function is always positive. This approach has recently been
applied by Di Paola et al. [5] in which the log-pdf was expanded as a Taylor
series. For the examples considered there, which included various first order
systems and the Duffing oscillator, it was shown that this technique gave good
agreement over the main body of the pdf, but no results for the tails of the
distribution were reported. A disadvantage of the adopted Taylor series solution
method is that it can not be used to analyse systems which contain non-linearities
whose derivatives are discontinuous at the origin (e.g. x\x\).
In this paper an approach similar to that proposed by Di Paola is used in which
an orthogonal expansion for the log-pdf is obtained. More specifically, a Hermite
polynomial expansion of the log-pdf is obtained by using a weighted-residual
approach to solve the governing differential equation for the log-pdf. The main
advantage of using this approach compared to a Taylor series solution is that it
is reasonably easy to incorporate non-linearities such as x|x| into the analysis.
The accuracy of the proposed numerical solution procedure to predict the
response statistics of various first and second order systems is investigated by
comparison with analytic results or numerical simulation.
2. THE FPK EQUATION AND LOG-PDF EQUATION
Many references (for example [6]) give details of the derivation of the Fokker-
Planck-Kolmogorov (FPK) equation for the determination of the transient
probability density function (pdf) of non-linear systems subjected to random
excitation. Here, the FPK is simply stated and then used to derive the associated
1582
log-pdf equation.
In order to apply the FPK equation it is necessary to write the equations of
motion of the system in state-space notation as follows:
z = g(z)-^Aw, (1)
where z is a vector containing the displacements and velocities of the system, A
is a square matrix (assumed here to be constant), g(z) is a general vector function
of the variables z, and is a vector of uncorrelated Gaussian white noise
processes, each having a spectral density of unity.
If z constitutes a Markov process, the FPK equation which governs the
transitional probability density function p(z,t) is given by [6];
dp(z,0 _
dt
' dz.dz
(2)
where n is the dimension of z and JS is a matrix given by:
B=2tiAA (2)
As mentioned in the introduction, equation (2) can be solved numerically using
the Galerkin method, the Finite Element method and the path-integral technique.
In order to obtain an equation which is more suitable for determining the tails of
the response distribution (and hence extremes) it is convenient here to express the
probability density function (pdf) as follows:
p(z,0=exp(p(z,0), (4)
where p is the log-pdf, i.e.:
p(z,0=ln;?(z,0-
Differentiating equation (5) with respect to t gives:
ap(z,0 _ 1 dpjzd) (^5^
dt piz,t) dr
Using equations (4) and (6), equation (2) can be re-written in terms of the log-pdf
p(z,t) as follows:
1583
dt ,=i
dz. ^‘dz.
I 1 ^ W
/ / = 1 ;=1
0p 6p
dz .dz . dz .dz.
(7)
Equation (7) is a non-linear partial differential equation governing the transient
log-pdf, and forms the basis of the technique proposed here. In what follows a
weighted residual procedure for solving this equation is presented.
3 WEIGHTED RESIDUAL SOLUTION FOR THE LOG-PDF
The weighted residual solution of equation (7) is considered for a first order
system and then for a second order system.
3.1 First Order Systems
For the case of first order systems equation (7) can be written as follows:
0^p(z,r)
f ap(z,oy
dt dz dz
dz'^
[ dz J
In the weighted residual approach adopted here the log-pdf and the system non¬
linearity are expressed as the sum of a finite number of weighted Hermite
polynomials, such that:
m
p(z,f)=EA.,(/)H,(z),
/=0
(9,10)
where
f g(z)ff^(z)e
(11)
and Hi{z) is the f th Hermite polynomial.
Substituting equations (9) and (10) into equation (8) gives:
1584
(12)
Multiplying equation (12) by /f*exp(-zV2)// 27i:, integrating over z from -«> to ~
and using the properties of the Hermite polynomials given in Appendix A gives:
k\\ = ~k\ i “ 1 ^
1=0 /=o
yt=0,
(13)
1=0 y=o
where /[zj, A;] is an integral of a triple Hermite product which, using the properties
given in Appendix A, may be shown to be given by:
npM-jlT.
{
i
!
' UL-k
I 2 2 2j
[ 2 2 2)
where k\s the smallest of ij and k.
(14)
Equation (13) represents a set of first order, non-linear, differential equations
which can be solved numerically for X*(0 using standard integration techniques.
The calculated values can then be substituted back into equation (9) to yield an
expression for p{z). To ensure that this is a valid pdf, the coefficient is
modified to ensure that the normalisation condition is satisfied.
For the situation when the response of the system is stationary (ie. dA,k/dt=0 for
all k), equation (13) reduces to a set of non-linear algebraic equations. Although
the solution of these equations can, in principle, be calculated using standard
numerical methods, these methods require a good first estimate of the solution in
order to achieve a good rate of convergence. For the systems considered in the
numerical examples section, the stationary pdf is calculated by integrating
equation (12) until stationary conditions are achieved.
This section has shown how a weighted residual approach can be used to obtain
a numerical solution to the log-pdf equation for first order systems. In what
1585
follows this technique is extended to deal with second order systems.
3.2 Second order systems
An identical procedure to that presented for first order systems is now applied to
a general second order system.
Let the log-pdf be expressed as follows:
n m
p(z,.z,,/) (15)
p=l q = i
and the system non-linearity be expressed as:
E (>6)
1=0 «=0
where
1
27r/!m!
and Hj is the z’th Hermite polynomial.
A set of non-linear, first order differential equations in terms of can be
obtained by substituting equations (15) and (16) into equation (7), multiplying by
Hp(z i)H^{z2)Qxp{~{z^^ +Z2^)/2)/(27c), integrating over 2, and from -<» to and
making use of equation (14). This yields an equation of the form:
^M^E^A+EE^taV/-’ (18)
ij ij l.m
where a,y and 6 j,/„,are constant coefficients dependent upon the system considered.
As in section 3.1, equation (18) can be solved numerically for Xp^ and then the
log-pdf can be obtained from equation (15). This procedure is used in the
following section to calculate the response statistics of some example systems.
In addition, it may be noted that the extension of the weighted residual technique
to multi-dimensional systems is straight-forward.
1586
4 NUMERICAL EXAMPLES
The techniques described above are used here to determine the stationary
response statistics of: i) first order systems and ii) second order systems. In all
cases the initial response conditions are assumed to have a Normal distribution
and the stationary response statistics are obtained by integrating the equations
governing the Hermite coefficients until stationary conditions are obtained. The
response pdf can then be obtained by substituting the Hermite expansion for the
log-pdf into equation (4) and applying the normalisation condition (to ensure that
it is a valid probability density function).
4.1 First Order Systems
Two first order systems are considered here. The first consists of a system with
a polynomial non-linearity, while the second consists of a drag non-linearity. In
both cases the results obtained using the proposed technique are compared with
the exact analytic solutions available, and both the pdf and log-pdf of the
response (z) are plotted.
4.1.1 Polynomial non-linearity
The first order system considered here has the following equation of motion:
x = -'^ax‘+M (19)
where a,=0.1, ^2=0.0, ^3=0.05, af=0.01, and/(0 is a zero mean, Gaussian,
white-noise excitation with a constant spectral value K=1/'k. The stationary
distribution in this case is given by:
5
p(x) =Aqxp(-2 '^^) (2^)
where A is the normalisation coefficient.
The distribution of x for this case is shown in Figure 1, where Figure la) shows
the pdf and Figure lb) shows the log-pdf In these figures a comparison with the
exact result, given by equation (20), is made with the results obtained using the
proposed technique with m=4 and m=6 in equation (9), and an equivalent
Gaussian distribution. In both cases it is seen that the response statistics are
highly non-Gaussian, and the proposed technique gives better agreement with the
exact result with m=6 than with m=4. Further, for the case when mi 6 it is found
that the proposed technique agrees exactly with the exact result. In many respects
1587
this agreement obtained using the proposed technique is not surprising since the
stationary pdf can be expressed exactly in the assumed form when ms 6.
However, there is no guarantee that the transient response can be expressed in this
assumed form.
4. 1 .2 Drag non-linearity
The system considered here has the following equation of motion:
x--x\x\+f{t), (21)
where f{t) is a zero mean, Gaussian, white-noise excitation with a constant
spectral value K=\!tz. The exact distribution in this case is given by:
p(x)=^exp(^-^), ‘ (22)
where A is the normalisation coefficient.
Unlike the system considered in section 4.1.1, the drag non-linearity and log-pdf
can not be expressed exactly using Hermite polynomials, and is therefore a more
difficult system to analyse using the proposed technique.
Figure 2 shows the statistical distribution of x for this situation, where Figure 2a)
shows the pdf and Figure 2b) shows the log-pdf In these figures a comparison
with the exact result, given by equation (22), is made with the results obtained
using the proposed technique with m=3 and m=7 in equation (9), and an
equivalent Gaussian distribution. It is seen that the response statistics are highly
non-Gaussian and that using an increased number of Hermite polynomials to
represent the log-pdf gives improved agreement with the exact result, especially
at the “tails” of the distribution. This is not surprising since the technique
proposed here deals with the log-pdf and is therefore well-suited to determining
the tails of the distribution.
4.2 Second Order Systems
From a practical point of view it is the response statistics of second (and higher)
order systems which are of vital importance to design engineers. In order to
demonstrate the performance of the proposed technique, two second order
systems are considered. In each case results are obtained for the displacement (^i)
pdf and the velocity (zj) pdf, and the results are plotted as a pdfs and log-pdfs.
4.2.1 Duffing Oscillator
The equation of motion considered in this case is given by:
1588
(23)
where / (t) is a zero mean, Gaussian, white-noise excitation with a constant
spectral value K=1/ti. In this case it is well-known [6] that the displacement and
velocity are statistically independent, and exact analytic results are available for
the displacement and velocity pdf s, such that:
p(z,)=C,exp(-^z,^-iz,‘'), p(z2)=C,exp(-izj\ (24,25)
where Ci and C2 are normalisation constants.
The stationary response statistics for this case are shown in Figure 3, where the
analytic results have been compared with those obtained using the proposed
technique and an equivalent Gaussian distribution. In this case it is found that the
proposed technique gives identical results for the jpdf (and hence displacement
and velocity statistics) provided that a sufficient number of Hermite polynomials
are used (i.e. n^4 and m^S in equation (15)). In some respects this is not
surprising since it is known that the stationary jpdf can be expressed exactly in
the assumed form given by equation (15), provided a sufficient number of terms
are used. However, as in previous examples, it is unlikely that the transient
response statistics can be expressed exactly in this form.
4.2.2 Duffing Oscillator with cubic damping non-linearity
The equation of motion considered in this case is given by:
z^ = -z2, = (26)
where f {t) is a zero mean, Gaussian, white-noise excitation with a constant
spectral value ^^=1/71:. Although there is no exact analytic result available for the
displacement distribution in this case, it is shown in Appendix B that there exists
an analytic result for the velocity statistics, such that:
piz^)=Atxp(-—-j^),
(27)
where A is the normalisation constant.
The stationary response statistics for this case are shown in Figure 4, where a
1589
comparison is made with numerical simulation and exact results. In the proposed
representation n=4 and m=8 are used in equation (15), and an equivalent
Gaussian distribution is shown for comparison. Up to the highest levels obtained
by simulation, the proposed method gives excellent agreement.
5 CONCLUSIONS
A numerical method for solving the associated FPK equation governing the
response statistics of non-linear systems subjected to random vibration has been
presented. The main advantage of this method over alternative techniques is that
it ensures that the tails of the response distribution remain positive, suggesting
that it well-suited to extreme value prediction. For the numerical examples
considered this was shown to be the case, in which excellent agreement with
exact analytic solutions and numerical simulation were obtained, especially at the
tails of the distribution. Although, in principle, the procedure developed is
applicable to an ^2-dimensional system, the computer costs increase rapidly with
n. However, the remarkable increases in computer performance over the past
decade suggests that the present procedure may, in the near-future, become a
viable option for calculating the response statistics of higher order systems.
REFERENCES
1. Wen, Y.K. Approximate method for nonlinear random vibration.
Proceedings of the American Society of Civil Engineers, Journal of the
Engineering Mechanics Division, 1975, 101,389-401.
2. Langley, R.S. A finite element method for the statistics of non-linear
random vibration. Journal of Sound and Vibration, 1985, 101(1), 41-54.
3. Naess, A. and Johnsen, J.M. Response statistics of nonlinear, compliant
offshore structures by the path integral solution method. Probabilistic
Engineering Mechanics, 1993, 8, 91-106.
4. Roberts, J.B. A stochastic theory for nonlinear ship rolling in irregular
seas, Journal of Ship Research, 1982, 26, 229-245.
5. Di Paola, M., Ricciardi, G., and Vasta, M. A method for the probabilistic
analysis of nonlinear systems, Probabilistic Engineering Mechanics,
1995, 10(1), 1-10.
6. Lin, Y.K. Probabilistic Theory of Structural Dynamics, McGraw-Hill,
New York, 1967.
1590
APPENDIX A - PROPERTIES OF HERMITE POLYNOMIALS
The following properties of the set of Hermite polynomials are used in section 3.
^ dz
=0 «=0
-n\
(Al)
dj^
dz
=nH^_^iz).
(A2)
APPENDIX B - ANALYTIC SOLUTION FOR THE VELOCITY
STATISTICS OF A PARTICULAR CLASS OF SECOND ORDER
SYSTEM.
For the second order systems considered in the numerical examples section the
associated FPK equation (equation (2)) can be written as follows:
dp{z^,z^) _ Bp{z^,z^ dg{z^,z^p{z^,z^
dt ^ 5z, dz^ dz'}
(Bl)
where
(B2)
Substituting equation (B2) into equation (Bl) and integrating over Zj from -oo to
gives:
dt ~
dz^
/
5p(2,,0 ,
az^^
(B3)
The first term on the right hand side of equation (B3) will reduce to zero since for
1591
it is reasonable to assume that as the displacement (z,) increases (or decreases)
without bound p{z^,z^ will tend to zero.
Provided that is continuous, the third term on the right hand side of
equation (B3) can be re-written as:
■■ — —
(B4)
The term in square brackets on the right hand side of this equation will reduce to
zero provided that g, is an odd function and p{z^,Zj) is symmetric with respect to
Zi for all values of This condition is satisfied by the systems considered in
sections 4.2.1, 4.2.2 when the response is stationary (i.e. 6p/9t=0). Consequently,
for the examples considered in the numerical examples section the third term
appearing in equation (B3) reduces to zero provided that the term on the left
hand-side is set to zero (i.e. stationary conditions).
Making use of the above observations and re-arranging, equation (B3) can be
written as:
(B5)
5^2 dzl
This is the associated stationary FPK equation for the first order system whose
equation of motion is:
(B6)
Thus, it has been shown that the stationary distribution Zi (alone) is identical to
that of a one-dimensional Markov process. Further, the solution of equation (B5)
can be written as;
2 f
p{zf, =A exp( j gfz)dz),
(B7)
where A is the normalisation constant. Consequently, the velocity statistics of the
second order systems considered in the numerical examples section can be
obtained analytically. This result is used to validate the accuracy of the results
in the numerial examples section.
1592
Normalised displacement pdf
Figure 1 . The stationary response statistics of the system given by equation (19).
— , exact solution; - • - , proposed technique (m=4); . , equivalent Gaussian
distribution, (a) displacement pdf, (b) displacement log-pdf (Note: the exact
solution and proposed solution (m=6) agree exactly in this case^
Figure 2. The stationary response statistics for the system given by equation (21):
— , exact solution; - • - , proposed technique (m=3); • • proposed technique
(m=7); ••••• , equivalent Gaussian distribution, (a) displacement pdf, (b)
displacement log-pdf
1593
Normalised velocity pdf Normalised displacement pdf
Figure 3a)
Figure 3b)
Normalised displacement Normalised displacement
Figure 3c) Figure 3d)
Figure 3. The stationary response statistics for the Duffing Oscillator given by
equation (23); - , exact solution; - • - , proposed technique; . , equivalent
Gaussian distribution, (a) displacement pdf, (b) displacement log-pdf, (c) velocity
pdf, (d) velocity log-pdf. (Note: the exact solution and proposed solution agree
exactly in this case.)
1594
Normalised displacement pdf
Figure 4c)
Figure 4d)
Figure 4. The stationary response statistics for the Duffing Oscillator with cubic
damping (see equation (26)); - , numerical simulation ((a) and (b) only); - •
proposed technique; - - , exact solution ((c) and (d) only); ••••• , equivalent
Gaussian distribution, (a) displacement pdf, (b) displacement log-pdf, (c) velocity
pdf, (d) velocity log-pdf.
1595
On the use of Finite Element solutions of the FPK equation
for non-linear stochastic oscillator response statistics
M. Ghanbari* and J. F. Dunne
School of Engineering, University of Sussex,
Palmer, Brighton, U.K. BNl 9QT.
SUMMARY
Numerical solutions of the stationary Fokker-Planck equation are obtained via
Langley's FE method and applied to non-linear oscillators driven externally by
Gaussian white-noise. A SDOF model, appropriate to large amplitude random
vibration of a clamped-clamped beam excited by band-limited noise, is used to
demonstrate convergence of predicted response marginal density functions and
extreme-value exceedance probabilities. Experimental verification ol this beam
model is shown at marginal density level using calibrated FEM-FPK
predictions and measured data. Convergence of a 4D FEM-FPK version is also
demonstrated at marginal density level, corresponding to a pair of non-linear
osciUators. This 4D method is appUed up to a practical storage limitation
imposed using a systematic FEM node numbering scheme.
LO INTRODUCTION
Finite element analysis has now found widespread application in the field of
stochastic dynamics for mathematical modelling of structures exposed to
random type loading. Applications vary from wave induced response
prediction for fixed and compliant offshore structures, seismic response
modelling of building structures during earthquakes, wind buffeting problems in
aerodynamics, and of course for modeUing the effects of roughness on road and
rail vehicular motions [1][2].
Predictions of reliability, in the form of appropriate response
statistics, can usually be obtained for linear structures from a finite element
structural model, combined with linear statistical and normal distribution theory
[3]. In cases where significant nonlinearities are present, accurate response
statistics cannot be obtained with linear theory, justifying alternatives such as
Monte Carlo simulation in the time-domain. Simulations can be very effective
in obtaining low-level statistics, for example the first few response moments
(even when there are many degrees-of-freedom). But to obtain highly accurate
information in the tails of the marginal response amplitude probability density
iunction (pdf), or to obtain accurate low-level extreme-value exceedance
probabilities, conventional simulations are impractical owing to the enormously
Ions runs needed to obtain confident probability estimates. Response amplitude
proWbilities are useful for initial reliability assessment, whereas extreme- value
■ Now at: RHP BEARINGS Ltd, 01dend.s Lan^,|^(^ehouse, Gloucestershire. GLIO 3RH UK.
statistics are very important for quantifying the likelihood of dangerously high
amplitudes being reached within a specified period of time, since these can
often be related directly to risks elsewhere.
Focusing initially on extreme-value statistics, a number of
vibration type approaches have been developed using a combination of
vibration theory, control techniques and simulation [4-8]. These methods are,
under certain conditions, capable of producing highly accurate results, although
for broad-band excitation, there is still a need for a direct, accurate, and
efficient approach which does not involve complicated intermediate stages or
simulation. More generally, within the theory of stochastic processes, there are
several important asymptotic routes to extreme- value prediction appropriate for
high response amplitude levels or long sample path durations. Local maxima
statistics for example, can be used to construct extreme-value statistics,
assuming independence of specified response maxima [9] - an . approach
frequently used for linear structures with Gaussian responses. Another
approach [8] uses the asymptotically Poisson character of local maxima at high
levels. An alternative route, via threshold crossing statistics, assumes
independence of up-crossings, by ignoring ‘bandwidth’ [5], an effect related to
the concentration of energy on response sample paths within a narrow band of
frequencies.
Threshold crossing statistics can be approached via the joint pdf
for a response amplitude process and its first derivative, which in turn can be
obtained for a general class of non-linear systems via Markov process theory
using the stationary Fokker-Planck-Kolmogorov (FPK) equation [2] [10]. This
equation can often be constructed directly from the equations of motion for a
stochastic dynamic system, provided the excitation sources can be modelled as
broad-band. Although few exact FPK solutions are known, in the past 30 years
there have been several types of numerical method proposed (references [11-
16] are just some examples of functional- type solutions, not to mention finite
difference methods). A good numerical solution opens-up possibilities for
obtaining accurate tails of the response amplitude pdf, and accurate extreme-
value statistics, thereby avoiding the need for enormously long simulations.
Whilst numerical FPK solution methods obviously differ in accuracy and
efficiency, there is one school of thought to suggest that finite element
structural modelling, followed by FE solution of the FPK equation, offers an
attractive and unified approach to structural reliability assessment.
But the accuracy of FEM-FPK predictions depend on the use
of a sufficient number of elements in the approximation, and to achieve a
certain level of accuracy, this is affected by the dimension of the problem, the
strength of nonlinearity in the model, and on the probability levels considered.
Computer storage limitations may also prevent the required number of elements
from being used. Furthermore stationary FPK solutions contain no ‘bandwidth’
information at the sample level. Therefore, the assumption of independent up-
crossings, intrinsic to all FPK based extreme- value predictions, must be
1598
checked using an alternative such as Monte Carlo simulation. But to justify
such lengthy checking, a realistic non-Hnear model is appropriate, for example
a large amphtude model for clamped-clamped beam vibration excited by broad¬
band noise [17].
In this paper, a SDOF beam model, and a pair of non-hnear
oscillators, are used to demonstrate use of Langley’s FEM-FPK method. In
both models, the parameters are known, and only external Gaussian white-
noise excitations are assumed. For the SDOF (beam) model, the rate of
convergence of the response amplitude pdf via the FEM, is demonstrated as,
more nodes are used. FEM-FPK based extreme exceedance probabilities are
then compared with long Monte Carlo simulations to show convergence with
increasing nodes, and to confirm the iusensitivity of bandwidth effects at the
probabihty levels considered. The model is justified at the pdf level, by showing
calibrated FEM-FPK predictions and experimental measurements. A 4D FEM-
FPK version is then applied to a pair of non-linear oscillators. This is to
establish the level of convergence reached in the response amphtude pdf, with
practical storage limitations imposed using a systematic FEM (global) node
numbering system.
2. The FPK Equation and Stationary Response statistics
When the force vector components arising in a structural dynamic model can be
treated as broad-band Gaussian processes, Markov process theory allows the
(forward) transition probabihty density function to be modehed using the
Fokker-Planck (FPK) equation (and backward transitions via the Backward
equation) [2]. From these densities important response statistics can be
obtained for use in rehabihty assessment. Under certain conditions, statistical
properties of response trajectories cease to vary with time aUowing a simpler
(stationary) FPK equation to be constructed. The starting point for
constructing the stationary FPK [10] is to express the system of (non-hnear)
equations of motion:
MX + F{X,X) = f{t) (1)
in appropriate state space fonn as follows:
i = ^(^) + Aw(0 (2)
where z represents an nxl vector Markov response process, w(0 is an
assumed nxl vector of zero-mean uncorrelated Gaussian white-noise
processes, whose spectral densities are unity, scaled by a constant square
matrix A, and g is an nxl vector of (non-hnear) system functions. The
stationary FPK equation associated with equation (2), can be obtained by
1599
identifying so called drift and diffusion coefficients [10] and by focusing on
steady-state trajectories. This leads to the partial differential equation:
(3)
where 5 = 2;&4A^and p(z) is the stationary joint probability density function
(jpdf) associated with the solution trajectories of the dynamic model. Boundary
conditions take a variety of forms [2], in addition to the normalisation condition
requiring that the total probability must sum to one. Very few exact solutions
to equation (3) are known which is one important reason why numerical
methods are increasingly used.
2.1 Marginal Densities and Extreme- Value Statistics
From the stationary FPK solution p(z), marginal density functions, and
corresponding moments, can be obtained by successive integration. The joint
bi-variate pdf for a response and its first derivative, can be used to obtain the
mean threshold crossing rate [10]. If this is accurate, then it offers an
approximate route to the distribution of extreme-values for response process
Zi . The required bivariate density function is in general obtained by integration
as follows:
+<«
and from equation (4), the mean up crossing rate is determined from:
4«>
= j Zi+l/’(UT.Z.+l)*W (5)
0
where u^ refers to the level of interest, and dz corresponds to the reduced
state space in which variables z, and z^+i do not appear. The Poisson
assumption of independent upcrossings [6] gives the approximate extreme-
value distribution for z,- (t) in the form:
■Fm(Ut) = prob{M(T) < %} = (6)
where M is the largest value of response process z,- (0 in the interval T. From
this, the exceedance probability for extreme-value is defined as:
1600
P{M(T) > u^}=1-F(Ut)
(7)
3. Finite Element Solution of the Stationary FPK Equation
There have been several finite element FPK solution methods proposed such as
the methods of Langley[12], Bergman[14], Spencer and Bergman[16] (also
applied to the Backward equation). These methods are essentially special
applications of the weighted residual (WRM) Galerkin-type approximation,
using shape functions defined over a fimte, rather than infimte region. The
Finite element method, for example proposed by Langley, has the additional
advantage of being able to deal efficiently with spatial dependence in the FPK
equation - a feature which does not normally occur in general structural
analysis problems. We do not propose to examine these methods in any detail,
except to give a brief outline of Langley’s method in the usual context of the
WRM appropriate for a stationary multi-dimensional FPK equation since this is
used later.
In Langley’s method [12], general use is made of piece-wise
shape functions to approximate, within a number finite elements, both the
solution P(z), and the non-linear appearing in equation (3). These
elements are usually rectangular for 2D, cuboid for 3D etc., and the total
number of nodes for a single n-dimensional element is m = 2" • The solution
P{z) over the entire domain is approximated in terms of the (unknown) nodal
values of the pdf as follows:
(8)
Shape functions N, (2) are chosen to take unit values at a specified node, and
zero at other nodes withm a particular element. For nodes which do not fall
within a particular element, the shape function values are zero.
In trial solution (8), the usual far-field boundary conditions are
modified to account for use of a finite region, namely that p{z) 0 when
(2) — > ±a where a is some large finite value. Substitution of trial solution (8)
for all elements over the entire finite region, allows the FPK equation residual
to be formed as follows:
^ i=l ;=1 CfZi f=l OZi
(9)
Equation (9) gives a measure of how well the entire solution p{z) satisfies
equation (2) and if the shape functions are chosen correctly, one hopes that the
1601
residual will reduce uniformly to zero as the number of elements is increased.
Weighted residuals are formed as follows:
j j R{^w{£)dz = 0
(10)
to provide the basis for selecting the nodal values of the approximate solution.
The weight functions w(z) , are chosen appropriately to be the same nodal
shape functions within those elements in which a particular node of
interest falls. For all other elements, the weight functions are taken to be zero.
This converts the weighted residual statement (10) to a finite number of
summations over all elements, which in principle generates the same number of
(linear) equations as unknown nodal probability values. This (generally sparse)
system of linear equations is in fact singular, but a unique solution can be
obtained by imposing the normalisation condition:
+«
lp(.l)dg = l (11)
To make best use of finite elements, the simplest type of shape function is used
to remove much of the burden in organising the FE code. The simplest choice
is indeed a piece-wise linear (Lagrange) function, where n-dimensional shape
functions can be constructed using appropriate multiples of the one-dimensional
version. However, were a trial solution to be attempted using these Lagrange-
type functions directly in equation (10), the FPK equation being second order,
would cause the first term to vanish since linear shape functions have only Cg
continuity. This problem was overcome [12] by integrating equation (10), to
generate the weak form of weighted residual statement for the FPK equation:
X ^ J ^ I (2)p(s)^[w(z)](iz = 0
j=l
i=l R
(12)
Equation(12) applied over aU the finite elements will yield a system of linear
equations for p. which can be obtained uniquely by imposing the normalisation
condition (11). As mentioned, far-field boundary conditions are applied at a
finite, rather than an infinite boundary, determined in practice using equivalent
linearization. This boundary is chosen at 5 or 6 standard deviations in aU
directions where the probability density outside this region is assumed to be
zero. The non-linear functions g;(2) appearing in equation (3) can also, for
reasons of computational efficiency, be approximated within a particular
element using the same shape functions as follows:
m
g(z)=J,gm) (13)
/=i
1602
This has the advantage of reducing the amount of pre-processing required to
handle the spatial variability in the FPK equation and is therefore useful where
parameter or model variations are intended. Once p(z) is calculated in the
above manner, the extreme- value exceedance distribution can be approximated
from equations (4)-(7).
A storage problem however arises in the use of the FEM, since
the linear coefficient matrix is of dimension x , where N is the number
of nodes. This obviously becomes excessive for anything more than the 2-
dimensional problem. Iterative solution techniques [18] can be used to solve
linear equations without need to store the coefficient matrix. But these methods
are suited to sparse matrices with narrow bandwidth, such as arise in structural
FEM - the best known being the Frontal solution method. Unfortunately the
benefits of using space saving methods in the FEM-FPK are to some extent
lost, as it can be shown that even if the theoretical bandwidth is no more than
3^^ , the practical bandwidth is significantly more than this when a systematic
nodal numbering scheme is used. Therefore for large n (say > 3), unless an
extremely sophisticated global node numbering system is used, the bandwidth
grows rapidly with the node number and the advantages of space saving
methods are not realised. For small n (e.g. n=l or 2) storage requirements are
not so critical, so a good solution can be obtained using standard sparse
system solution techniques.
4. Application to SDOF Model
The first application of this FEM-FPK is to a SDOF oscillator model:
Z + 2^a),Z + a,ZZ~ + o:,z|z| + colZ + k^Z^ = Aw{t) (14)
which represents a system with both hnear and non-linear damping, and linear
plus cubic stiffness. The excitation w(0 is a unit intensity stationary zero mean
Gaussian white noise process, scaled to required level by parameter A. Some
evidence will be shown shortly to confirm this as a realistic vibration model for
one dimensional large amplitude vibration of clamped-clamped beam, at least
in the tails of the response amplitude marginal probability density function [17].
First we intend to show convergence of predicted FEM-FPK pdf solutions with
increasing nodes; and then extreme exceedance predictions based on the FEM-
FPK using equations (3)-(7), in both cases predictions are compared with
Monte Carlo simulations - this includes an assessment of the Poisson
assumption using equation(5). To complete the section, pdf predictions using a
calibrated form of model (14), are compared with experimental measurements
of clamped-clamped beam vibration.
Putting oscillator model (14) into state space fonn, gives a two-
state vector Markov model:
1603
(15)
Z2
-a^z2Zf -a2Z2lz2l~0)Ui
"1
’0 0*
“ 0 ■
dt -j-
_0 A_
db{t)_
and the corresponding stationary FPK equation (3) can then be written as:
M^-^-^(zi/(z)) + -^((2(?a)„Z2 + «iZ2zf + +^3^1 )/(^)) - ^
UZ2 ^Z\ 0^2
... (16)
subject to far-field and normalisation boundary conditions as described in
section 2. The parameter values used for the model are :
^ = 0.0, = 0M2,a, = 0.015 , = 144.34,^ = 3021 , A = 200 . Application
of statistical linearization gives =0.0138 and simulation -reveals a
eq
bandwidth parameter value £ = 0.97 . Note although expKcit parameter values
are given here, only damping to intensity ratios appear in the FPK solution,
which confirms that bandwidth effects cannot be accounted for, hence the need
for an independent check on the accuracy of extreme-value predictions.
Figure 1. : Response amplitude marginal density function for SDOF model via FEM-FPK
solutions and simulation.
To check FEM-FPK predictions, conventional time-domain
simulations are used here. These involve three stages : 1) construction of
1604
excitation sample paths ; 2) numerical integration, using a standard 4th order
Runge-Kutta scheme; and 3) post-processing, involving transient removal to
create a number of output sample paths for estimation of marginal response
densities, and for creation of sections of length T, for use in estimation of
extreme exceedance probabilities. A truncated Whitaker filter [19] is used to
allow convergence of numerical integration using white-noise excitation
samples assembled at discrete time intervals of duration At , giving a Nyquist
bandwidth f,, = . In the use of the Whitaker filter At is fixed, aUowing
interpolation to smaller time steps At, producing rapid reduction of the
truncation error.
Figure 1. shows the response marginal pdf obtained via FEM-
FPK and simulation. Using symmetry over 1/4 region, the number of nodes in
the FEM is increased as shown, from 9 nodes to a total of 225. This number
does not create computational or storage problems using a systematic FEM
node numbering scheme, and convergence has clearly occurred by 225 nodes
Figure 2. shows extreme exceedance probabilities using FEM-
FPK predictions via the threshold crossing rate equation (5) in equations (6)
and (7), compared with very long Monte Carlo simulations. Two durations
have been examined; T=1 second, as shown in figure 2a. ; and T=100 seconds
in figure 2b. In both figures, convergence of FEM-FPK predictions are
compared with converging simulation results for reducing At in the Whitaker
filter. Note that the simulations seem to have converged for
At = 0.001 seconds , and the FEM-FPK predictions seem to similarly converge
but only when a total of 961 nodes is used. Results for 225 nodes, which are
perfectly adequate for marginal density predictions as shown in figure 1, are
clearly not accurate for extreme-value statistics. The final level of agreement
between simulation and FEM-FPK predictions would suggest that the Poisson
assumption does indeed hold below probabilities of 10 ^ justifying use of
equation(6).
The statistical variability m extreme exceedance estimates p is
predicted using the ratio cr. / p = l/ Np , where cr * is the standard deviation
in p and N is the sample size. For low probabilities, a large value N=1000
extreme-values was needed for each duration, giving confident estimates of p
above 10'^. The CPU time needed here for the simulations varied between 30
and 300 times the FEM-FPK requirement.
Figure 3 shows FEM-FPK predictions, based on a calibrated
SDOF model (14), via the parameter estimation method [20], compared with
direct measurements of large amplitude beam vibrations [17]. This experimental
clamped-clamped beam rig was Im in length, and 25mm by 3mm section, with
density p = 7850kg /m^ and Young’s Modulus E = 190GN/m^ Band-
limited white noise excitation was applied using a shaker positioned near one
1605
Figure 2. : Extreme Exceedance probabilities for SDOF model. FEM-FPK predictions
compared with Monte Carlo simulations: a) T=1 second; b) T=100 seconds.
- Simulated Response (delta t=0.004 sec) - Simulated Response (delta t=0.002 sec)
. Simulated Response (delta t=0.001 sec) —A— F.E 15x15=225 nodes
RE 21x21=441 nodes -X- F.E 31x31=961 nodes
end of the beam, the central ampHtiide being measured with an accelerometer.
Full details of the rig, instrumentation, and data processing procedure can be
1606
0.5
Figure 3. : Response amplitude marginal density functions for beam model. Calibrated SDOF
model showing FEM-FPK predictions and experimental measurements: a) linear
model; b) non-linear model.
found in [17]. For this beam, linear theory gives the first three small amplitude
natural frequencies as 15Hz, 42Hz, and 82Hz respectively, whereas measured
large amplitude responses are concentrated around 25Hz in a single-degree-
of-freedom type motion without any observed responses above this bandwidth.
Figure 3a shows a calibrated linear model, in contrast to figure 3b, which
compares a calibrated version of equation (14). The magnitudes of the raw
parameter estimates obtained using the moment method [20] are given as :
<J/A' =5.09xl0■^ ajA^ =18.26xl0■^ =0.47x1 0"^ =132.56,
1607
and = 2322 . Note, the intensity level A need not be known for FEM-FPK
predictions at the same excitation level as the measured data. This is
advantageous since excitation may in general be difficult to measure.
5. Application to a pair of non-linear oscillators
A 4D FEM-FPK version can be applied for example to an isolating suspension
system [21], which in general represents a pair of coupled non-linear equations:
X + od -h K^x + ‘¥yxO^ =
0 -f- 156 + G^O + G-^6^ -f* y 6x^ — ^2^2 (0
By setting Zi=x, Z2=0 and Z3 = x, Z4 = 0 the state space model becomes:
Z3 ^
ro
0
0
0 ^
0 ^
-azs
-K^Zi -K^zl -Jz^zl
+
0
A
0
A 2
W; (f)
^3
Z4
0
0
0
0
0
~ (JjZt “^3^2 ^yZ'iZ^ J
lo
Ail
0
^2 )
where A^ and A2 are white noise intensities; Aj2and A^^ are cross-correlations,
all scaled by a factor In: . The FPK equation can be written as follows:
2nA{ -f- 2nA, + ;rA2 — {z^p) - — {z^p)
dzl *“ dzsdz^ “ ^4
^4
6z^
+^[(a^3 +^1^1 + K.x +n,zl)p]+-^[{i5z, + g,z2 + G2Z2 +rz,zl)p\ = 0
C^3 OZ4
... (19)
Equation (19) can be solved exactly in specific cases, such as when:
A A
Ai2 = A21 = 0,— = -^ = 27 (a nonzero constant), the solution is then:
a p
1 fl
1.
1
1
1
p(^i,Z2,^»Z4)~C©qpj^ +^^Zi ■{^Qz2+^Qz2'^~^y2:iZ2| 2^ ^3 2A
... (20)
To establish the finite region for application of the 4D FEM-FPK, the
equivalent linear fonns of equations (17) are:
a
x + ax + {K^ + 2>K^a\ +y(jl)x = A^a:^ {t)
6 + j5B + {G^ +3G3(Tg +yal)B = A^w^_{t)
(21)
Solving (21) for the amplitude and velocity mean-square values leads to:
1608
<7! =
^ 0 Ml
and = — -
a
(22) & (23)
(^1 =
- - r- and (jI
l5(.G,+3G,<Ji+r<yl) "
,2 _ ^Ai
a
^ (24) & (25)
To demonstrate this 4D FEM-FPK, the parameter values were set to:
if, = Gi = 1.0 and with r = 0.0and A^=A^=\l4n. Note,
although introducing no change to the FEM-FPK solution method, setting
7 = 0.0 effectively uncouples equations (17) - which can in fact then be solved
with a 2D FPK [10]. But from the FEM viewpoint the problem can still be
Figure 4. : Response amplitude marginal density function for pair of nonlinear oscillators.
4D FEM-FPK predictions compared with exact solution.
seen as if it were 4D, the advantage being that exact marginal densities can be
trivially obtained from the exact jpdf. Use of statistical linearisation [1], gives
(after 1 1 iterations) converged values of = cr^ = 0.538 which are used,
along with symmetry, to define the finite FE region:
0<zi <3.7,0<z, <5.0,0<z3 <3.7,0<z4<5.0. ^ made-up of
4D equivalents of cuboid elements, each with 16 nodes. Figure 4 shows
response amplitude marginal density predictions obtained with this 4D FEM-
1609
FPK compared with the exact solution. Only two cases are considered,
namely 3*^ = 81 and 6^ = 1296 nodes respectively. This second case involves
solution of a linear system of equations in 1296 unknowns. But since a
systematic node numbering system is used, the bandwidth of the coefficient
matrix is very much larger than the minimum value of 81. Although the
comparison for the second case, shows good agreement, very many more nodes
would clearly be needed to obtain accurate extreme- value statistics. This would
not be possible with the present approach owing to computer storage
limitations.
6. Conclusions
Accurate stationary response amplitude pdfs are shown to be efficiently
obtained using a 2D FEM-FPK approach applied to a realistic non-linear model
for SDOF clamped-clamped beam vibration. This model has been corroborated
experimentally at the marginal density level. Application of the FEM to
extreme-value prediction, via threshold crossing statistics, shows good
agreement compared with simulation, for a beam type model. Moreover,
application of a 4D FEM-FPK associated with a pair of non-linear oscillators,
shows reasonable agreement with the exact solution at the marginal level. But
for n > 2, use of a systematic FEM node numbering scheme is not suitable for
extreme- value prediction owing to the large bandwidth of the FEM coefficient
matrices. To circumvent this problem for application to higher dimensions a
sophisticated global node numbering scheme is needed to enable space saving
linear equation solution techniques to be of benefit.
References
1. Roberts, J.B. and Spanos, P.D., Random Vibration and Statistical
Linearization, Wiley, Chichester, 1990.
2. Lin, Y.K. and Cai, G.Q., Probabilistic Structural Dynamics, Mc-Graw-
Hill. 1995.
3. Leadbetter, M.R., Lindgren, G. and Rootzen, H., Extremes and related
properties of random sequences and processes. New York : Springer
- Verlag 1983.
4. Winterstein, S.R. and Ness, O.B., Hennite moment analysis of non-linear
random vibration, in Computational Mechanics of Probabilistic and
Reliability Analysis, Lausanne; Ehne Press, 1989, Chapter 21, pp. 452-
478.
5. Naess, A., Approximate first-passage and extremes of narrow-band
Gaussian and Non-Gaussian random vibration. Journal of Sound and
Vibration, 1990, 138, pp. 365-380.
6. Naess, A. Galeazzi, F. and Dogliani, M., Extreme Response predictions
of non-linear compliant offshore structures by stochastic linearization .
Applied Ocean Research, 1992,14, pp. 71-81.
1610
7. Winterstein, S. R. and Torhaug, R., Extreme response of Jack-up
structures from limited Time-Domain Simulation. Proceedings of the 12th
International Conference of the 0 MAE- ASMS, 1993, Vol 2, 251-258.
8. Dunne, J.F., An optimal control approach to extreme local maxima for
stochastic Duffing-type oscillators. Journal of Sound and Vibration, 1996,
193(3), pp. 597-629.
9. Naess, A., On a rational approach to extreme value analysis: Technical
nolQ. Applied Ocean Research, 1984 ,6(3), pp. 173-174.
10. Soong, T.T. , Random Differential Equations in Science and Engineering,
Academic Press - New York, 1973.
1 1 . Bhandari, R.G. and Sherrer, R.E., , Random vibration in discrete non
-linear dynamic systems. Journal of Mechanical Engineering Science,
1968, 10, pp. 168-174.
12. Langley, R.S., A finite element method for the statistics of non-linear
random vibration. Journal of Sound and Vibration, 1985, 101(1), pp. 41
-54.
13. Soize, C., Steady state solution of Fokker-Planck equation for higher
dimensions. Probabilistic Engineering Mechanics, 1989, 3, pp. 196-206.
14. Bergman, L.A., Numerical solution of the first passage problem in
stochastic structural dynamics, in Computational Mechanics of
Probabilistic and Reliability Analysis; Lausanne; Elme Press 1989, pp.
479-508.
15. Kunert, A., Efficient numerical solution of multidimensional Fokker-
Planck equations associated with chaotic and non-linear random
vibrations. Vibration Analysis - Analytical and Computational, ASME
DE-57, 199 l,pp. 57-60.
16. Spencer, Jr. B.F. and Bergman, L.A., On the numerical solution of the
Fokker Planck equation for nonlinear stochastic system. Non-linear
Dynamics, 1993, Vol 4 , pp. 357-372.
17. Ghanbari, M., Extreme response prediction for random vibration of a
clamped- clamped beam, 1996, D.Phil. Dissertation - University of
Sussex.
18. Press W.H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W.T.,
Numerical Recipes - the art of scientific computing, Cambridge
University Press, 1989.
19. Oppenheim, B.W. and Wilson, P. A., Continuous digital simulation of
the second order slowly varying drift force. Journal of Ship Research,
1980, 24(3), pp. 181-189.
20. Roberts, J.B., Dunne, J.F. and Debonos, A., Parameter estimation for
Randomly excited non-linear systems, in lUTAM symposium on Advances
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Mechanical Engineering Science, 1960, 2, pp. 195-201.
1611
1612
RANDOM VIBRATION II
SIMULATION OF NONLINEAR RANDOM VIBRATIONS
USING ARTIFICIAL NEURAL NETWORKS
Thomas L. Paez*
Susan Tucker
Chris O’ Gorman
Sandia National Laboratories
Albuquerque, New Mexico, USA
Abstract
The simulation of mechanical system random vibrations is important in
structural dynamics, but it is particularly difficult when the system under
consideration is nonlinear. Artificial neural networks provide a useful tool for
the modeling of nonlinear systems, however, such modeling may be inefficient
or insufficiently accurate when the system under consideration is complex.
This paper shows that there are several transformations that can be used to
uncouple and simplify the components of motion of a complex nonline^
system, thereby making its modeling and random vibration simulation, via
component modeling with artificial neural networks, a much simpler problem.
A numerical example is presented.
Introduction
Structural system random vibration simulations are required in a wide
variety of applications. Development of techniques that can generate such
simulations accurately and efficiently is important, particularly in frameworks
where numerous simulations are required, frameworks tike Monte Carlo
analysis. In practically all situations where the excitation is Gaussian and the
system under consideration is nonlinear, the responses will be nonlinear and
non-Gaussian, and it is important that simulations preserve the characteristics
of the response as accurately as possible.
Artificial neural networks (ANNs) have been applied to the autoregressive
modeling of nonlinear system random vibrations. Investigations have shown
that nonlinear structures can be modeled with ANNs, at least in the case of
simple systems. (See, for example, Yamamoto, [15].) In principle,
complicated systems can also be mc^eled using ANNs. This can be done
directly (i.e., without any substantial transformation of the input or output
data) using many types of ANNs. As the complexity of the system increases,
an ANN that can naturally and efficiently accommodate a large number of
inputs must be used for system simulation. When a mechanical system is
modeled using an autoregressive ANN to directly simulate motions at a large
* This work was supported by the United States Department of Energy under
contract No. DE-AC04-94AL85000. Sandia is a multiprogram laboratory
operated by Sandia Corporation, a Lockheed Martin Company, for the United
States Department of Energy.
1613
number of degrees of freedom, a very large number of exemplars of motion
will be required to train the ANN to accurately represent the system. The
reason is that it takes a large number of exemplars to adequately populate a
high dimensional input space.
This paper shows how the ANN modeling of nonlinear structures can be
made more efficient and accurate when using data measured during
experimental, stationary random vibration. There are a number of operations
that can be performed on the data to accomplish these goals. Among tiiese are:
(1) principal component analysis, (2) localized modal filtering, (3) elimination
of statistically dependent components of motion, and (4) transformation of the
components of motion to statistically independent, standard normal random
signals. These operations are briefly describe in the following sections, along
with the modeling of the components with two types of ANN - the feed
forward back propagation network (BPN) and the connectionist normalized
linear spline (CNLS) network. An example is included to assess the random
vibration simulation capabilities of the ANNs. The accuracy of the simulations
is evaluated in terms of spectral and probabilistic measures.
Data Reduction
It is important to reduce the dimensions of motion of a complex system for
the reason listed in the introduction, i.e., the amount of data required to train a
very complex system directly is great. Further, because the CNLS net is a local
approximation network, it is important to minimize the number of network
inputs. The reason is tiiat network size grows rapidly with the number of
inputs. To limit the complexity of the input/ouiput mappings required to model
a complex system, the system motions can be decomposed into simple
components. In general, the ANN modeling of physical systems can often be
made more efficient and accurate by preprocessing the training data using ^y
of a number of simplifying transformations. Among these are: (1) principi
component analysis, (2) localized modal filtering, (3) elimination of statistically
dependent components of motion, and (4) transformation of the components of
motion to statistically independent, standard normal random signals. The hopes
in using these transformations are that the ANN required to model a component
of behavior will be simpler than a model for the entire system, and that a
simpler model will be easier to train. Exactly how these transformations fit into
the response simulation framework will be discussed more later, the overall
framework is described in Figure 4. These operations are briefly described in
the following subsections.
Principal Component Analysis - SVD
Principal component analysis of complex structural system motions is
aimed at decomposing the motions into their essential constituent parts. A
special example of this is the modal decomposition of linear systems, and
analogous decompositions can be defined for nonlinear systems using, for
example, singular value decomposition (SVD), or a principal component
analysis ANN.
1614
SVD is described in detail, for example, in Golub and Van Loan [4]. It can
be used to decompose linear or nonlinear structural motions in the following
way. Let AT be an nxN matrix representing the motion of a structural system
at N transducer locations and at n consecutive times. The form of the SVD is
X = UWV^ = uwv^ (1)
V is an V X V matrix; its columns describe the characteristic shapes present in
the rows of AT. TV is an V x V diagonal matrix whose nonnegative elements
characterize the amplitudes of the corresponding shapes in V. The elements in
TV are normally arranged in descending order. Its largest elements correspond
to the most important components in the representation, is an nxN matrix;
its columns are filtered versions of the motions represented by the columns in
X. The reason that U is said to be a filtered form of the motions in X is that
the columns of both V and U are orthonormal with respect to themselves.
Therefore,
U = XVW~^
(2)
and T^TV ^ serves as a filtering coefficient. Because some of the elements of TV
may be zero or nearly zero, indicating components that do not contribute
substantially to the characterization of X, the elements of TV ^ are taken as the
inverses of the diagonal elements in TV that are greater than a cutoff level; zero
or near zero elements in TV are replaced with zeros in TV”V The approximate
equality on the right side of Eq. (1) indicates that some components of the
representation can be zeroed, and still maintain a good approximation to A" . In
the experimental framework, the comjwnents of TV whose ratio to the
maximum value is lower than the experiment noise- to-signal ratio are set to
zero. The matrices «, w, and v are the matrices I/, TV, and V with
components removed.
The columns of the matrix u are the principal components of the
representation. It is the evolution of the system represented in the columns of u
which we seek to simulate. Once models are established to simulate system
response through simulation of the columns of the models can be used
along with system initial conditions to predict structural response. The
predicted response can be used, along with Eq. (1), to synthesize response
predictions in the original measurement space.
Principal Component Analysis - ANN
ANNs can be used in a number of application frameworks, and one of the
focuses of this paper is to show how components of a complex system motion
can be modeled with ANNs. However, a particular ANN can also be used as a
means for decomposing complex system motions into simpler components.
This ANN is the principal component ANN (PCANN). (Baldi and Hornik,
[2]) The PCANN is simply a multi-layer network of perceptrons like the
standard BPN (Freeman and Skapura, [3]), but it has a particular geometry,
1615
shown in Figure 1. Let (xy) be a row vector of N elements, the jth row of data
in the matrix X defined above. Then the collection of all the rows of AT provide
n exemplars - both input and output - for training the PCAIW in Figure 1.
Some important features of the ANN in Figure 1 are that (1) it is a BPN with
one hidden layer, and (2) die number of neurons, R, in the hidden layer is
smaller than N, ±e number of columns in the measurement matrix. The idea
behind the PCANN is that it compresses the information in the input layer into
the information present in the hidden layer, then uses this information to
reconstruct, as accurately as possible, the original signal on the output layer.
To obtain optimal effect from the PCANN, sigmoidal activation functions
would normally be used in the hidden and output layer neurons. However,
when linear activation functions are used in tiie hidden and output layer
neurons, the ANN weights are related to the components of the SVD, Eq. (1).
Figure 1. Geometry of the principal component artificial neural network.
Let the jth row of m be the hidden layer outputs, in Figure 1. The
columns of u are the principal components of the PCANN representation. It is
the evolution of the system represented in the columns of u which we seek to
simulate. Once models are established to simulate system response through
simulation of the columns of ii, the models can be used along with system
initial conditions to predict stmctural response. The predicted response can be
used, along with the portion of the PCANN to the right of the hidden layer, to
synthesize response predictions in the original measurement space.
Normally, only one of the principal component analyses described in this
and the previous sections would be applied to the data.
Modal Filtering
Modal decomposition of complex system motions is often used for the
simplification of mechanical system response when the model for the system is
assumed linear or approximately linear. In fact though, data-based modal
1616
decomposition can be used on any collection of data; its purpose is to break the
into simple narrowband components, thereby simplifying system
characterization, and perhaps simulation. When such a decomposition is used
on nonlinear system or general system data, it is often referred to as modal
filtering.
Modal filtering of measured data can be performed in one of two
frameworks. First, a single modal analysis can be used to filter measured da^
into its component parts. The problem with this is that when a system is
nonlinear, its characteristics can change with response magnitude, and the
number of principal components (modes) also changes. It is difficult for a
single model to accurately capture such changes. The second alternative is to
specify multiple modal filters, and use each one m a particular range of
response amplitudes. A very simple realization of this type of analysis would
involve the use of two modal models. One would be applied to data below a
particular threshold, and the other would be applied to data above the
threshold. A more complicated application creates a linear modal model at each
step in a system analysis. Such a model is described in Hunter [7].
The form of a modal filter is similar to the SVD, but the means for
obtaining the filter factors is much different. Meirovitch [9] describes the
theoretical operations included in ±e definition and use of a modal filter.
AUemang and Brown [1] outline practical means for performing data-based
modal analysis. The form of the modal filter can be expressed as
X = US<I>^ = us^'^ (3)
where X is the same mechanical system motion representation as above. The
columns of (p represent the characteristic shapes of the system, and the diagonal
matrix S contains normalizing factors. When X comes from a line^ system the
columns of U are the linear modal components of the system motion. As with
the SVD, the approximate equality on the right indicates that some components
of the representation can be eliminated, and still maintain a good approximation
to X. It is the evolution of the columns in u (a reduced form of U) which we
seek to simulate. Once models are established to simulate system response
through simulation of the columns of m, the models can be used along with
system initial conditions to predict structural response. The predicted response
can be used, along with Eq. (3), to synthesize response predictions in the
original measurement space.
When multiple decompositions are used to filter the motions of a complex
system, then multiple expressions like Eq. (3) are used to obtain modal
components.
Elimination of Statistically Dependent Components
Although some of the principal component analyses of the previous
sections may produce orthogonal components, some of the components may
be completely or highly statistically dependent upon others. For example, a
structure may have two modes with nearly the same modal frequency. One
1617
modal motion may be nearly a sine wave, and the other may be nearly a cosine.
The motions are practically orthogonal, but they are still statistically dependent.
Statistical independence of sources implies orthogonality, but orthogonality
does not necessarily imply statistical independence. For the sake of efficiency,
we seek to eliminate statistically dependent components from the set to be
modeled, then reintroduce these components during physical system
simulation. In this way, ANN modeling of structural behavior is simplifi^.
When a dependency exists, it can be characterized using the conditional
expected value of the variables in one column of u given values in another
column of m. This requires approximation of a joint probability density
function (pdf) of the data, and this can be obtained using the kernel density
estimator. (See Silverman, [13].) The pdf approximation is known as the
kernel density estimator (kde). Let (wj), j = denote the row vectors of
the matrix u. The kde of the random source u is given by
/«(«) =
— oo < M < oo
(4)
where a is an N x 1 variate vector, K{.) is a kernel function, and is a
window width parameter of the kernel function. The kernel function can be my
standard probability density function, and often the pdf of a multivariate
standard normal random vector is used. That is
where a: is the N x 1 variate vector. Using the kde in Eq. (4), the estimator for
the conditional pdf of elements in one column of u given the values in another
column of u can be obtained using the standard formulas. (See, for example,
Papoulis [11].)
A statistical dependency between two columns of u can be detected by
forming the bivariate pdf estimator of the random source of the two columns
using the kde with data from the columns in question, then evaluating and
plotting the conditional expected value of one variable, given a range of values
of the other variable. At each point where the conditional expected value is
evaluated, the conditional variance can also be evaluated. The conditional
expected value and variance can be evaluated for situations in which the data in
the two columns of u are lagged with respect to one another. If a lag is found
where the conditional variance is uniformly small, i.e., small at all locations
defined by the conditioning variable, then a statistical dependency has been
detected, and the functional form of the dependency is defined by the
conditional expected value. The dependent variable can be eliminated from
modeling consideration. When modeling has been completed and it is
necessary to restore the eliminated component, this can be accomplished using
the conditional expected value develops here.
1618
The effect of eliminating components of motion that are completely
dependent on other components is to eliminate some columns in the matrix u.
Denote the reduced matrix Uj. \ our objective is to model the evolution of the
columns of Uj. with an ANN.
Rosenblatt Transform
The previous step produces a description of the motion of a complex
structure in terms of a set of components, none of which is completely
statistically dependent on others. We can further transform the components, tiie
columns of into signals that are statistically independent with Gaussian
distributions. The transformation that accomplishes this is the Rosenblatt
transform. (See Rosenblatt, [12].) The Rosenblatt transform has the following
form.
Z2 = ^”'(^“2tel(“2'«l))
where the z^-, i = 1,...,N, are uncorrelated, standard normal random variables,
0(.) is the cumulative distribution function (cdf) of a standard normal random
variable, is its inverse, and the F(.) are the estimated marginal and
conditional cdf’s of the random variables that are the sources of the colunms of
u. These approximate cdf’s can be obtained by integrating Eq. (4), and this can
be accomplished directly when the kernel used in Eq. (4) is Eq. (5).
The Rosenblatt transformation is uniquely invertible because the exact and
approximate cdfs used in Eq. (6) are monotone increasing. The ckta in the
matrix w can be transformed to the standard normal space by using it in Eq. (6)
in place of the a’s. The matrix z is composed of the elements z/, / = 1,...,A^ ,
and is the same size as the matrix with the same number of nonzero
columns.
It is the evolution of the values in these columns that is to be simulated with
ANNs. Because the columns in z are statistically independent, we need only to
create ANN models for signals in individui columns. Once models are
established to simulate system response through sirnulation of the columns of
z, the models can be used along with system initial conditions to predict
structural response. The predicted response can be used, along with the inverse
form of Eqs. (6), to synthesize response predictions in the original
measurement space.
1619
Modeling of Component Motion with ANNs
Our ultimate objective is to simulate complex system motion, and we aim to
do this by simulating the components of system motion obtained using the
decompositions and transformations described above. Many ANNs are suitable
for this task. The two that we consider in this paper are the feed forward back
propagation network (BPN) and the connectionist normalized linear spline
(CNLS) network. The BPN is the most widely used ANN and it is described
in detail in many texts and papers, for example Freeman and Skapura [3], and
Haykin [5]. The BPN is very general in the sense Aat it can approximate
mappings of relatively low or very high input dimension. It has b^n shown
that, given sufficient training data, a BPN with at least one hidden layer and
sufficient neurons can approximate a mapping to arbitrary accuracy (Homik,
Stinchcombe, and White, [6]).
The CNLS network is an extension of the radial basis function neural
network (Moody and Darken, [10]). It is described in detail in Jones, et.al.,
[8]. It is designed to approximate a functional mapping by superirriposing the
effects of basis functions that approximate the mapping in local regions of the
input space. Because it is a local approximation neural network, we cannot use
the CNLS network to accurately approximate mappings involving a large
number of inputs. The CNLS network has not been widely used for the
simulation of oscillatory system behavior.
To simulate a column in z using either of the ANNs described above, we
configure the net in an autoregressive framework. This configuration uses as
inputs previous response values and the independent excitation, and yields on
output, the current response. Figure 2 shows such an application
schematically. The quantity zji denotes an element in the ith column of z at the
jth time index. The quantity qj denotes the excitation at the jth time index. Lj
denotes a lag index. There are m system response input terras; there are M+2
excitation terms The configuration shown in Figure 2 implies our belief that
there is a mapping
^j+U ~
and that the ANNs can identify that mapping. The subscript i on the function
g(.) indicates that the functional mapping varies from one column of z to the
next, and a different ANN models each mapping. It is normally anticipated that
the time increment, At, separating system motion measurements that are the
rows of the matrix X is small relative to the period of motion of the highest
frequency component we intend to simulate.
We seek to train both types of ANN to model the behavior of the
oscillations represented in the columns of z. One ANN of each type (BPN and
CNLS net) is used to model each column of z. The inputs to the ANN are
current and lagged (past) values of the transformed response and the one-step-
into-the-future value, the current value, and lagged values of the excitation.
The ANN output is the transformed response one step in the future. Both the
1620
ANNs are trained using the scheme described in Figure 3. The ANN inputs
are transformed using a feed forward operation. The ANN output is compared
to the desired output, and the error is used to modify the ANN parameters.^ 'Die
BPN uses a back propagation and gradient descent scheme in each trainmg
step. The CNLS network uses least mean square (LMS) plus random sampling
scheme to identify its parameters. The desired effect of training in both types of
ANN is to modify the parameters of the network to diminish the error of
representation of the input/output mapping.
^J-k4
^i-T i
J
qj+l
Qj-^
A
N
N
Figure 2. Schematic of ANN in autoregressive configuration.
Figure 3. Schematic describing training sequence for ANNs.
Summary
Figure 4 summarizes the decomposition, simulation, and modeling of
structural motion described in the previous sections. The principal component
analysis in the second box in the top row refers to one of the following: SVD,
PCANN, or modal filtering. The synthesis in the fourth box in the second row
refers to the corresponding inverse operation - Eq. (1), the right half of Figure
1, or Eq. (3).
1621
In the example that follows, one form of principal component analysis will
be combined with ANN simulations to model a nonlinear structure’s random
vibrations.
Decomposition and Modeling Operations
Simulation of Response
Figure 4. Summary of operations in system simulation.
Numerical Example
We simulate in this example the motion of a simple, nonlinear 10 degree-
of-freedom system excited with a Gaussian white noise. Figure 5 shows a
schematic of the system. The damping connecting the masses is linear viscous.
The springs have a restoring force that is a tangent function. The system
physical parameters are summarized in Table 1. Training data for the neural
networks were generated by computing response over 8192 time steps (box
number one on the top line in Figure 4); excitation and responses at ten
locations were recorded. The time increment between response realizations is
0.04 second. Figure 6 shows the displacement response at the 10th mass. This
is the location where the simulation-to-experiment comparisons are made in the
present example, and where the simulation yielded the poorest match to the
experimental results.
The responses were placed in a matrix X as referred to in the previous
sections, and its SVD was computed (box number two on the top line of
Figure 4). The singular values of the response indicate that accuracy of about
89% should be achieved by simulating the system response with its first four
components. The four components were not strongly statistically dependent,
so none was removed. The kde of the four components indicate that none is
highly non-Gaussian, therefore, the Rosenblatt transform was not used. The
first four components of the response were modeled with both BPN and
CNLS nets (box number five on the top line in Figure 4).
The entire system was tested in autoregressive operation, as described in
Figure 2, using data generated over 1000 steps of response computation. The
initial conditions and excitation were used to start then execute a random
vibration response simulation with ANNs in the space of u (box number one
on the second line of Figure 4). The test was iterated, i.e., the estimated
responses at step j were used as initial values for response predictions at time
indices greater tiian j. The first and most significant column of u from the test
1622
data and the first column from the ANN simulated data are compared in
Figures 7a and 7b. This is the dominant component of the response. The match
is good, particularly in view of the fact that the simulation is iterated. Note that
although these and later response predictions remain fairly well in phase with
the test responses, this is not usually the case. Typically, we hope that the
simulated response amplitudes match the test responses well, and accept the
fact that phase will usually be lost. Figures 8a and 8b compare the spectral
densities of the signals shown in Figures 7a and 7b. (Tliese were comput^
using the technique described, for example, in Wirsching, Paez, and Ortiz,
[14].) A block size of 256 data points was used, along with a Hanning
window, and an overlap factor of 0.55. The first harmonic of motion, at 0.25
Hz, is very well match^. A third harmonic of motion appears to be present in
botii the test and simulated signals; the BPN provides a better match of the
third harmonic than the CNLS net. Figures 9a and 9b compare the kde’s of the
signals shown in Figures 7a and 7b. The responses are clearly non-Gaussian,
as anticipated because of nonlinearity, and to some extent the simulated
component responses match the character of the test response. This match
needs to be improved to maximize the quality of the simulation. (Some of the
mismatch is caused by the limited data - 1000 points - upon which the
comparison is based.) However, as will be seen, the simulated synthesized
responses have kde’s that match test response kde’s quite well.
Table 1. Parameters of the test system.
Tangent
Maximum
Viscous
Index
Mass
Stiffness
Deformation
Damping
1
1.0
40
1.0
0.40
2
0.95
38
1.0
0.38
3
0.90
36
1.0
0.36
4
0.85
34
1,0
0.34
5
0.80
32
1.0
0.32
6
0.75
30
1.0
0.30
7
0.70
28
1.0
0.28
8
0.65
26
1.0
0.26
9
0.60
24
1.0
0.24
10
0.55
22
1.0
0.22
11
2
1.0
0.02
The simulated responses are now reconstructed by substituting into Eq. (1)
using the simulated u (box number four on the second line of Figure 4), and
the results are compared to the test response at mass 10 (box nuniber five on
the second line of Figure 4). The result from the BPN simulation is shown in
Figure 10a; the result from the CNLS net simulation is shown in Figure 10b.
Of course, the matches are quite good since the dominant component is well
simulated by both ANNs. Figures 11a and 11b compare the spectral densities
of the signals shown in Figures 10a and 10b. The spectral densities estimated
from the simulated signals match those of the test signals well up to the
frequency where components are no longer simulated; the CNLS net does a
slightly better job of matching the test spectral density than the BPN. At mass
10 the rms response of the test signal is 0.45 in, and the rms values of the
simulated responses are 0.47 in and 0.40 in for the BPN and CNLS net.
1623
respectively. The corresponding kde’s of test and sinaulated responses were
computed and are shown in Figures 12a and 12b. The probabilistic character of
the responses appears to be matched well by the ANN simulations.
q
Figure 5. A nonlinear spring-mass system.
Figures 6. Response at mass 10 to Gaussian white noise input.
Figures 7a and 7b. Comparison of test and ANN simulated responses -
Component 1. BPN simulation on left; CNLS simulation on right. ANN
simulation - solid line; test data - dashed line.
1624
10*’ 10° io’
Frequency, Hz
Figures 8a and 8b. Comparison of the spectral density estimates of test and
ANN simulated responses - Component 1. BPN simulation on left; CNLS net
simulation on right. ANN simulation - solid line; test data - dashed line.
Figures 9a and 9b. Comparison of the kernel density estimators of test and
ANN simulated responses - Component 1. BPN simulation on left; CNLS net
simulation on right. ANN simulation - solid line; test data - dashed line.
Figures 10a and 10b. Test (dashed) and simulated (solid) responses at a mass
10 in the simple system - BPN (left), CNLS net (right).
1625
10
10
Figures 11a and 11b. Test (dashed) and simulated (solid) estimated response
spectral densities at mass 10 in the simple system -• BPN (left), CNLS net
(right).
Figures 12a and 12b. Test (dashed) and simulated (solid) response kde’s at
mass 10 in the simple system - BPN (left), CNLS net (right).
Conclusions
A sequence of operations leading to the simulation of nonlinear structural
random vibrations with ANNs is described in this paper. Such simulations are
desirable because of their efficiency and relative accuracy. It is argued that if
the motions can be decomposed and transformed into simple components, then
the simulation will be simpler and more accurate. A numeric^ example
confirms that relatively simple motions can, indeed, be modeled with ANNs -
the BPN and CNLS net (a local approximation network) in particular. Given
sufficient training data accurate simulations of simple components should
always be possible, though obtaining satisfactory accuracy may r^uire
substantial effort. Accurate simulations should correctly reflect probabilistic,
spectral, and all other characteristics of the simulated component responses.
When component responses are correctly modeled then system level responses
will be simulated accurately.
1626
References
1. Allemang, R., Brown, D., (1988), “Experimental Modal Analysis,”
Chapter 21 in Shock and Vibration Handbook, Third Edition, Harris, C.,
editor, McGraw-Hill, New York,
2. Baldi, P., Homik, K., (1989), “Neural Networks and Principal
Component Analysis. Learning from Examples without Local Minima,”
Neural Networks, 2, 53-58,
3. Freeman, J., Skapura, D., (1991), Neural Networks, Algorithms,
Applications, and Programming Techniques, Addison-Wesley, Reading,
Massachusetts.
4. Golub, G. H., Van Loan, C. F., (1983), Matrix Computations, Johns
Hopkins University Press, Baltimore, Maryland.
5. Haykin, S., (1994), Neural Networks, A Comprehensive Foundation,
Prentice Hall, Upper Saddle River, New Jersey.
6. Homik, K., Stinchcombe, M., White, H., (1989), “Multilayer
Feedforward Networks are Universal Approximators,” Neural Networks,
V. 2, 359-366.
7. Hunter, N, (1992), “Application of Nonlinear Time Series Models to
Driven Systems,” Nonlinear Modeling and Forecasting, Casdagli, M.,
Eubank, S., eds., Santa Fe Institute, Addison-Wesley.
8. Jones, R. D., et. al., (1990), "Nonlinear Adaptive Networks: A Little
Theory, A Few Applications," Cognitive Modeling in System Control, The
Santa Fe Institute.
9. Meirovitch, L., (1971), Analytical Methods in Vibrations, The Macmillan
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10. Moody, J., Darken, C., (1989), “Fast Learning Networks of Locally-
Tuned Ifrocessing Units,” Neural Computation, V. 1, 281-294.
11. Papoulis, A., (1965), Probability, Random Variables, and Stochastic
Processes, McGraw-Hill, New York.
12. Rosenblatt, M., (1952), "Remarks on a Multivariate transformation,"
Annals of Mathematical Statistics, 23, 3, pp. 470-472.
13. Silverman, B. W.(1986), Density Estimation for Statistics and Data
Analysis, Chapman and Hall.
14. Wirsching, P., Paez, T„ Ortiz, K., (1995), Random Vibrations - Theory
and Practice, Wiley, New York.
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UK, pp. 475-486.
1627
1628
Dynamic Properties of Pseudoelastic Shape Memory Alloys
D.Z. and Z.C. Feng*’
“Department of Engineering Mechanics
Hunan University, China
^’Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139 USA
ABSTRACT
In this paper, we report a set of vibration transmission experiments that are conducted to investigate how the
pseudoelasticity of shape memory alloys (SMAs) affects the transmissibility characteristics of a spring-mass system,
where a shape memory alloy rod is used as a spring. The tests are conducted by subjecting the SMA bar under
tension-compression and under bending. The test results indicate that compared with ordinary alloys, SMAs have a
much higher damping. Most importantly, the damping property depends on the amplitude of the responses indicating
that the spring-mass system is nonlinear. Furthermore, the high damping property persists to the high frequency
limit (above 1 KHz) permitted by the equipment setup.
Keywords: Shape memory alloys, Nitinol, vibration damping, hysteresis
1. INTRODUCTION
Shape memory alloys (SMAs) such as Nickel-Titanium (NiTi) and Copper- Zinc-Aluminum (CuZnAl) exhibit nonlin¬
ear mechanical properties. Specifically, in a temperature environment which is higher than the phase transformation
temperature of the material, when an applied stress exceeds a certain threshold, stress induced phase transformation
generates large strains in the material so that the material can accommodate large strain with little change in the
applied load. On the unloading cycle, a reverse phase transformation takes place, thus no permanent deformation
remains. Moreover, the stress-strain relationship during this loading and unloading cycle shows hysteresis. This
phenomenon is called pseudoelasticity. The lower elastic modulus and higher material damping of SMAs are desir¬
able characteristics of passive vibration control systems. Previous investigations show that the damping of SMAs
displays a peak and the elastic modulus demonstrates a trough in the vicinity of the phase transformation during
the heating and cooling processes [l]-[3]. Some researchers have pointed out that the pseudoelasticity of SMAs can
augment passive damping significantly in structural systems and SMAs have potential application in passive vibra¬
tion control [4]-[5j. Recently, a research study has been undertaken to measure acceleration trainsmissibility of NiTi
shape memory alloy springs (6).
Among all of these studies listed above, the investigators failed to realize that the measurements on the damping
coefficient and Young’s modulus depend on the specific test methods used. In [7], we have used continuum models
of pseudoelastic SMA model to illustrate the nonlinear nature of the dynamic response of SMAs. In this paper, we
present experimental evidence on the nonlinear behavior of the materials.
When used as actuators, SMAs typically have a very slow response to thermal actuations unless SMAs are made
into very fine filaments. For this reason, SMAs have the perception of being limited to low frequency applications.
But when used as a passive vibration damper, the high damping property is not limited to low frequency applications.
Here, we demonstrate the high damping property of SMAs to a high frequency permitted by our equipment setup.
(Send correspondence to Z.C.F.)
Z.C.F.: Email: zfeng@mit.edu; Telephone: 617-253-6345; Fax: 617-258-5802
1629
Figure 1- The experimental setup of the tension-compression test.
2. EXPERIMENTAL SETUP AND PROCEDURE
Our vibration tests include tension-compression test and bending test. The experimental setup for these two types
of tests are similar. Fig. 1 is a schematic diagram of the tension-compression test setup. A test specimen is mounted
vertically by a clamping fixture attached to a shaker. Its top end is fixed to a mass. In bending test, the specimen is
fixed horizontally at one end and with a mass of ISOg at the other end. The base harmonic motion is generated from
HP3562A Dynamic Signal Analyzer, then amplified through B&K Power Amplifier 2707 ajad finally provided by B&K
Exciter body 4801 with general purpose head 4812. Both base excitation and the response at the end of a specimen
are picked up by two accelerometers Picomin 22 and PCB 309A respectively. The signals pass through two Kistler
5004 Charge Amplifiers to be feed into an HP 3562A Dynamic Signal Anal3^er to obtain transmissibility curves.
The transmissibility curves are easily acquired from the frequency response measurement by a sine sweep within the
frequency range of interest. During each measurement, the base motion level is kept constant and recorded in voltage
for convenience. Ail test specimens are circular rods, acting as springs in the testing mechanical spring-mass system.
The SMA rods are binary NiTi alloys obtmned from Shape Memory Applications Inc. of Santa Clara, California.
Steel and aluminum rods are also used as our specimens for comparison. For each specimen, the transmissibility
properties of the system are investigated under several base motion levels to observe the nonlinearity of the system
dynamics. All tests are carried out at room temperature.
3. TENSION-COMPRESSION VIBRATION TRANSMISSIBILITY EXPERIMENT
The parameters of specimens are listed in Table 1. The acceleration transmissibility curves obtained are shown in
Fig. 2-Fig. 5, where the mass in the spring-mass system is 250g. It is readily seen that the pseudoelasticity of SMAs
does have significant effects on vibration transmissibility characteristics. The NiTi rods behave as nonlinear softening
springs. When the base motion level increases, the resonant peak shifts to lower frequency and the peak becomes
lower. This indicates that the equivalent elastic modulus of the NiTi rod becomes lower and its material damping
becomes higher as the base motion level (measured by the input voltage to the shaker) increases. Higher base motion
levels result in larger strains in the rods. For example, the strain amplitudes in the NiTi #2 rod are about 0.001%,
1630
SMAOI
60; -
’ 0,3V
50 h
, - 2.0V
0 8.0V
600 650 700 750
Frequency Hz
Figure 2. Tension-compression vibration transmissibility curves of NiTi Rod # 1 at different base motion levels.
0.008%, 0.015% and 0.03% corresponding to the base motion levels 0.3V, 2.0V, 4.0V and 8.0V respectively. This
amplitude dependence of the elastic and damping properties of the material is significally different from those of
conventional alloys. Fig. 4 and Fig. 5 are the transmissibility curves of a steel rod and an aluminum rod. Under two
different base motion levels, the transmissibility remains the same. Since the system response at resonances is very
big owing to the very low damping of the materials, our experimental setup prevents us to test higher input levels.
Quantitative results related to the transmissibility curves are listed in Table 2.
The nonlinearity of the system dynamics is apparent in the amplitude dependence of the damping ratio and the
resonance frequency. This is seldom appreciated in the past. Although the damping ratio and the material Young’s
modulus are often measured using vibration tests, only one excitation level is used in many tests. Thus this nonlinear
phenomenon is often not noticed.
Table 1. Parameters of Test Specimens of
Tension- Compression Vibration
Specimen
Length (mm)
Diameter (mm)
Mass (g)
NiTi # 1
240
5.86
42.3
NiTi # 2
250
4.96
31.6
Steel
617
4.80
86.9
Aluminum
716
6.36
61.2
1631
Figure 3. Tension-compression vibration tfansmissibility curves of NiTi Rod # 2 at different base motion levels.
Frequency Hz
Figure 4. Tension-compression vibration transmissibility curves of steel rod at different base motion levels.
1632
60.
Aluminum
55r
50-
45 1-
CD I
"40[-
'3
S 35-
1
2 30 -
25-
20 1-
i
isi-
0.5V
2.0V
! i
I \
/ 1
/ \
500 550
Frequency Hz
Figure 5. Tension-compression vibration transmissibility curves of the aluminum rod at different base motion levels.
Table 2. Some Experimental Results of
Tension-Compression Vibration
Specimen
Base Motion
Level (V)
Resonant
Frequency (Hz)
Modal Damping
Ratio (%)
Resonant Peak
Value (dB)
NiTi # 1
0.3
684.2
0.52
39.6
2.0
673.0
0.83
35.5
8.0
650.0
1.30
30.9
NiTi # 2
0.3
613.0
0.55
38.8
2.0
596.5
1.00
32.1
4.0
58l0
1.10
29.2
8.0
551.9
1.60
25.6
Steel
0.5
714.6
0.09
55.9
2.0
714.5
0.10
54.3
Aluminum
0.5
533.3
0.12
55.6
2.0
533.2
0.10
5'5.9
Our results listed in Table 2 show that SMAs have higher dampings at a frequency above 500 Hz. This indicates
that the pseudoelastic properties of SMAs persist up to reasonably high frequencies relevant to vibration and acoustic
control. It is well known that the higher damping of the materials is associated to the movement of the boundaries
between different metallurgical phases. It is believed that such movement of interfaces takes place at the speed of
sound in the solid. Therefore, the higher damping properties of pseudoelastic SMAs are expected to persist up to
even higher frequencies.
To investigate the damping properties of pseudoelastic SMAs at even higher frequencies, we conducted another
set of tension-compression tests. To acquire higher resonant frequency of the system, we shorten the specimens and
use a smaller mass of 150g. The lengths of the specimens of NiTi # 1, NiTi # 2 and steel rods are 150mm,120mm and
230mm respectively. Fig. 6, Fig. 7 and Fig. 8 are their transmissibility curves measured. It is evident that the SMA
1633
40
SMA #1
Figure 6. Tension-compression vibration transmissibility curves of NiTi Rod # 1 at higher resonant frequency.
rods are effective as vibration isolators up to a frequency over IK Hz. The limited power of the shaker prevented us
from experimenting with even higher frequencies. Some quantitative measurements related to the transmissibility
curves are given in Table 3.
Table 3. Some Experimental Results of
Tension-Compression Vibration at Higher Frequency
Specimen
Base Motion
Level (V)
Resonant
Frequency (Hz)
Modal Damping
Ratio (%)
Resonant Peak
Value (dB)
NiTi # i
0.3
1028
0.23
39.6
1.0
1021
0.51
37.1
2.0
1011
0.53
34.8
4.0
997
1.10
32.4
NiTi # 2
0.3
1061
0.39
41.8
1.0
1052
0.68
37.7
2.0
1039
0.83
34.5
4.0
1016
0.96
31.0
Steel
0.3
1504
0.02
70.2
2.0
1503
0.01
70.9
Prom Table 2 and Table 3 we can see that the SMA specimens have damping ratios which are an order of magnitude
higher than those of steel specimens. When the SMA specimens are subjected to maximum base excitation delivered
by the shaker (which has 100 pound maximum force), the peak tension-compression strain remains less than one
percent. Since NiTi materials have been shown to have very good fatigue life if the strain is less than 2 percent [8],
it is thus safe to say that even higher damping ratio can be achieved in practical usage if the SMA component is
designed to operate at close to 2% peak strain. A 2% peak strain seems to be small, but it is nearly an order of
magnitude higher than the maximum elastic strain in many conventional metal alloys.
1634
■ 0.3 V
4- 1.0 V
40r 0 2.0 V
X 4.0 V
CO
^35[-
i X
g30l- X
Frequency (Hz)
Figure 7. Tension-compression vibration transmissibility curves of NiTi Rod# 2 at higher resonant frequency.
Steel
Frequency (Hz)
Figure 8. Tension-compression vibration transmissibility curves of the steel rod at higher resonant frequency.
1635
The tension-compression tests at large strains could be conducted using a larger shaker. We point out, however,
that in conducting such tests, lateral (bending) vibration must be prevented. Since the base motion is also a source of
parametric forcing of the bending modes, the tension-compression tests can be complicated by the onset of parametric
instability of the bending modes.
4. BENDING VIBRATION TRANSMISSIBILITY EXPERIMENT
To examine the damping properties of SMAs under larger strains, we conducted bending tests. In the bending
tests, the resonance frequency (the fundamental mode) is much lower than the tension-compression mode, thus the
shaker can exert a much larger displacement. The parameters of specimens are listed in Table 4. The acceleration
transmissibility curves are shown in Fig. 9 - Fig. 12. The same qualitative trend as the tension-compression
experiment is seen. As we would expect, the resonant peaks of the system with NiTi rods are more heavily suppressed
than the tension-compression tests. For example, the resonant peaks of the systems with specimen NiTi # 1 and
NiTi # 2 drop from 35.5 dB to 18.9 dB and from 32.1 dB to 14.4 dB respectively for the same base motion level of
2 V. Since strains generated in bending tests are larger than in tension-compression tests and larger strains lead to
larger areas of the hysteresis loop in the stress-strain curve of the NiTi rods, higher damping and lower equivalent
elastic modulus occur in the NiTi rod. This indicates that the pseudoelasticity of SMAs makes SMAs more effective
particularly in large- amplitude vibration controls. Unlike the tension-compression vibration tests, the acceleration
transmissibility curves for the steel and the aluminum rods at different levels of base motion no longer collapse to a
single one. This nonlinearity is not caused by the constitutive laws of the material but rather by the geometric and
inertial nonlinearity owing to the large amplitude of the vibration. In addition the compliance of the fixture may
also contribute Co the nonlinearity. Some quantitative measurements related to the transmissibility curves are given
in Table 5.
Table 4. Parameters of Test Specimens of Bending Vibration
Specimen
Length (mm)
Diameter (mm)
Mass (g)
NiTi # 1
100
5.86
17.6
NiTi # 2
80
4.97
10.1
Steel
200
4.80
28.2
Aluminum
r 150
6.36
12.8
Table 5. Some Experimental Results of Bending Vibration
Specimen
Base Motion
Level (V)
Resonant
Frequency (Hz)
Modal Damping
Ratio {%)
Resonant Peak
Value (dB)
NiTi # 1
0.1
24.8
2.2
28.0
0.3
24.1
2.9
26.0
1.0
22.4
4.4
20.9
2.0
20.9
4.6
18.9
NiTi # 2
0.1
22.8
3.5
23.2
0.3
21.9
4.5
20.3
1.0
19.8
6.8
16.3
2.0
18.2
7.1
14.4
0.1
22.2
0.3
0.34
43.8
Aluminum
0.1
23.1
0.097
47.3
0.3
23.0
0.17
46.0
Although larger peak strains can be achieved in the bending tests, peak strain occurs only at locations farthest
from the neutral axis. Perhaps the more cost effective way of utilizing SMAs in bending vibration control is to place
SMAs away from the neutral axis in a composite beam.
1636
25
Frequency (Hz)
Figure 10. Bending vibration transmissibility curves of NiTL Rod # 2 at different base motion levels
1637
15 20 25 30 35
Frequency (Hz)
Figure 11. Bending vibration transmissibility curves of the steel rod at different base motion levels
ALUMINUM
60 1 - ^ -
I
"inl-
Frequency (Hz)
Figure 12. Bending vibration transmissibility curves of the aluminum rod at different base motion levels
5. CONCLUSION
Our test results indicate that compared with conventional alloys, SMA materials have a much higher damping ratio.
In fact, the material properties are nonlinear. We observe that the resonance frequencies and the damping ratios
are dependent on the excitation levels. When conventional parameters such as the damping ratio and the Young’s
modulus are used to characterize these nonlinear materials, we must realize that these parameters will be dependent
on the test methods and procedures.
As we would expect based on the fact that the stress- induced phase transformation takes place at the speed
of the sound, the higher damping associated with pseudoelasticity persists to frequencies high enough for practical
vibration and acoustic applications.
BIBLIOGRAPHY
1. S. Ceresara, A. Giarda, G. Tiberi, F.M. Mazzolai, B. Coluzzi and A. Biscarini, Damping characteristics of
Cu*Zn-Al shape memory alloys. Journal de Physique IV, Vol.l, C4, 235-240, 1991
2. H.C. Lin, S.K. Wu and M.T. Yeh, Damping characteristics of TiNi shape memory alloys. Metallurgical Transac¬
tions A, Vol. 24 A, 2189-2194, 1993
3. C. Li and K.H. Wu, Systematic study of the damping characteristics of shape memory alloys. Smart Material,
SPIE Proceedings, Vol. 2189, 314-325, 1994
4. E.J. Graesser and F.A. Cozzarelli, Shape memory alloys as new materials for aseismic isolation. Journal of
Engineering Mechanics, Vol. 117, No. 11, 2590- 2608, 1991
5. P. Thomson, G.J. Balas and P.H. Leo, The use of shape memory alloys for passive structural damping. Smart
Mater. Struct., Vol. 4, 36-42, 1995
6. E.J. Graesser, Effect of intrinsic damping on vibration transmissibility of Nickel-Titanium shape memory alloy
springs. Metallurgical and Materials TVansactions A, Vol. 26A, 2791-2796, 1995
7. Z. C. Feng and D.Z. Li, Dynamics of a mechanical system with a shape memory alloy bar. J. Intell. Mater. Syst.
and Struct. Vol. 7, 399-410, 1996,
8. C. Liang and C.A. Rogers, Design of shape memory alloy springs with applications in vibration control. J. of
Vibration and Acoustics. Vol. 115, 129- 135, 1993.
1639
1640
INVESTIGATION OF THE REDUCTION IN THERMAL
DEFLECTION AND RANDOM RESPONSE OF COMPOSITE PLATES
AT ELEVATED TEMPERATURES USING SHAPE MEMORY
ALLOYS
Z. W. Zhong and Chuh Mei
Department of Aerospace Engineering
Old Dominion University, Norfolk, VA 23529-0247
ABSTRACT
The reduction in thermal deflection and random response of composite
plates with embedded shape memory alloy fibers (SMA) at. elevated
temperatures is investigated. The stress-strain relations are developed for a
thin composite lamina with embedded SMA fibers. Finite element equations
and computational procedures are presented for composite plates with
embedded SMA with the consideration of nonlinearities in geometry and
material properties. The results demonstrate that SMA can be effective in
reduction of thermal deflection and random response of composite plates at
elevated temperatures.
INTRODUCTION
For some alloys, a given plastic strain can be completely recovered
when the alloy is heated above the characteristic transformation (austenite
finish, Tf ) temperature. This shape memory effect phenomenon is attributed
to the material which undergoes a change in crystal structure knovm as a
reversible austenite to martensite phase transformation. The solid-solid phase
transformation also gives an increase in Young’s modulus by a factor of three
or four and an increase in yielding strength approximately ten times. The
transformation temperature can be altered by changing the composition of the
alloy. Many alloys are known to exhibit the shape memory effect. They
include the copper alloy family of Cu-Zn, Cu-Zn-X (X=Si, Sn, Al, Ga), Cu-
Al-Ni, Cu-Au-Zn and the alloys of Ag-Cd, Au-Cd, Ni-Al, Fe-Pt and others
[1]. Nickel-Titanium alloys (Ni-Ti or Nitinol) are the most common SMA [2].
SMA has been used as actuators for active control of buckling of
beams [3] and shape control of beams [4], It is also being investigated in
active vibration control of beams [5], rotating disk/shaft system [6], large
space structures [7] and flow-induced vibration [8]. The design of extended
bandwidth SMA actuators was also investigated [9].
Composites with embedded SMA fibers use shape memory alloy fibers
as reinforcements which can be stiffened or controlled by the addition of heat.
1641
The concept of active damage control of hybrid material systems using the
SMA as embedded induced strain actuators has been proposed [10]. Active
vibration control of flexible linkage mechanisms using SMA fiber-reinforced
composites has been investigated [11]. Acoustic transmission and radiation
control by use of the SMA hybrid composites was presented [12, 13].
Manufacturing of adaptive graphite/epoxy structures with embedded Nitinol
wires was recently reported [14]. Detailed formulations of the bending, modal
analysis and acoustic transmission of SMA reinforced composite plates have
been presented [15, 16].
A limited number of investigations on the thermal postbuckling
deflection of SMA fiber-reinforced composite plates exists in the literature. A
feasibility study on reduction of thermal buckling and postbuckling deflection
of composite plates with embedded SMA fibers was reported [17] and the
passive control of random vibrations of SMA embedded composite plates
using the analytical continuum method was presented [18]. The finite element
analysis of random response suppression of composite panels using SMA was
recently proposed [19]. In all those investigations, the material nonlinearities
were not considered.
In the present paper, the stress-strain relations for a thin composite
lamina with embedded SMA fibers are presented. Governing equations for
postbuckling and random vibration of SMA fiber-reinforced composite plates
are presented using the finite element method. Solution procedures using the
incremental and Newton-Raphson iteration methods are presented. Numerical
examples are given.
CONSTITUTIVE AND FINITE ELEMENT FORMULATION
Consider a thin composite lamina; for example, graphite-epoxy,
having an arbitrary orientation angle and with SMA fibers embedded in the
same direction as the graphite fibers. The stress-strain relations for such a
lamina in the principal material directions are derived in Appendix. For a
general k-th layer with an orientation angle 0, the stress-strain relations, Eqs.
A17 and A18, become
r - n
/
a.
1
' to
1 *
<
k
i
e
m
y
k
m ^
AT,T>T, (1)
and
f
a.*
A
Q*
k
<
> — -
<
♦
► AT
V
y^.
k
T<T.
(2)
1642
where [0*] and [Q]„are the transformed reduced stiffness matrices of the
composite lamina and the composite matrix, Ts is SMA stimulating
temperature (austenite start temperature), respectively. The resultant force
and moment vectors of the SMA fiber-reinforced composite plate are defined
as
(,{N},m) = Cjo},il,z)cb
or
PL
'A*
'T’W
1^“']
^ ^ r
iMj
B*
D^JIkJ I
iHl
[Kj
where the laminate stiffness [A*], [B*] and [D*] are all temperature
dependent, the recovery inplane force (A"*} and moment {m'} vectors are
dependent on temperature and prestrain (see Fig. A2). The vectors {N^} and
{M^} are the force and moment vectors due to initial stress {oo}. The
recovery and the thermal inplane forces and moments due to the SMA
recovery stress and the temperature change, respectively, are
({a;},{m;}) = J_ Na; T> T; andnuH T <T^
({jV^}.Kr}) = L“ [e Ar(U>&. T>T,
(4)
(5)
= e* a; AT{U)d^, T<T,
The inplane strain and curvature vectors, {s'^} and (k), are defined
from the von Karman strain-displacement relations as
W.x
{win
^ ^
;
.+Lj/2
> + <
• + z\
7^'
1
to
= K} + {e;} + «}+4K} (7)
= {s°}+z{k}
where wo is the initial deflection (sum of the converged thermal deflections
from previous incremental steps) which is necessary for the nonlinear material
1643
properties considered, and u and v are the in-plane displacements and w is the
transverse deflection measured from the initial deflection position.
The governing equations for a SMA fiber-reinforced composite plate
subjected to a combined thermal and random excitation loads can be derived
through the use of a variational principle. The system equations in finite
element expression can be written in the form
H 0
0
K, K,
Ks
01 rr oi 0
0 0 0 0 0 0
+
1
0"
1 '
_^mbo
+
- 1
i o
2
0 J
0
0
"^3
0
ojj
p.(0l
1
1 IP.]
\PL
1
1
1 0 J
L-* mtiT
r
’|p'
L mr ^
+
o
io
I
(8a)
or
[M\m + m - [K^,r]+[K]+ [K. ] + j +
= {^(0) +{p^)- {p:)- {p. } + {p^ro) - {p:o) - {p.o)
(8b)
where [M] is the system mass matrix, [K] and (P) are the system stiffness
matrices and the load vector, respectively; [K^] and are geometric
stiffness matrices due to the recovery stress a*( or {A*}) and thermal
inplane force vector respectively; [Ka] and [Ko] are the geometric
stiffness matrices due to the initial stress (cjo} and the initial deflection {Wo},
respectively; [Nl] and [N2] are the first and second-order nonlinear stiffness
matrices which depend linearly and quadratically upon displacement {W},
respectively; [Nlo] is the first-order nonlinear stiffness matrix which is
linearly dependent on the displacement (W) and the known initial deflection
(Wo}. The subscripts b and m denote bending (including rotations) and
membrane components, respectively; subscripts r, c and o represent that the
corresponding stiffness matrix or load vector is dependent on the recovery
stress a*, initial stress {oo} and initial deflection {Wo}, respectively; and the
subscripts B, Nm and NB indicate that the corresponding stiffness matrix is due
to the extension-bending coupling laminate stiffness [B], membrane force
vectors {Nm} (=[A*]{s;,}) and {Nb} (=[B*]{k}), respectively. The stiffness
matrices [K], [Nl], [Nlo] and [N2] are also temperature dependent and all the
matrices in Eq. (8b) are symmetric. Detailed derivations of the governing
equations and expressions for the element matrices and load vectors are
referred to [19-21].
1644
SOLUTION PROCEDURES
Equation (8b) is a set of nonlinear ordinary differential equation with
respect to time t, and some of the load vectors are independent to time t. The
solution for Eq, (8b) consists of a time-dependent solution and a time-
independent solution as
{W} = {WJ + {W(r)}, (9)
where {Ws} is the time-independent solution and its physical meaning is the
large thermal deflection, and {W(f)}j is the time-dependent dynamic solution
whose physical meaning is small random oscillations about the static
equilibrium deformed position {Ws}.
Thermal Deflection or Postbuckling
For nonlinear material properties of SMA, the incremental method
should be employed. This implies that the material properties are treated as
constant withhi each small increment of temperature. Substituting Eq. (9) into
the system equation of motion, Eq. (8b), and neglecting the higher order terms
of {W(r)}j for small random response, two sets of equations can be obtained.
One is the time-independent nonlinear algebraic equations which yield the
thermal postbuckling deflection (Ws) and it can be written as
= {P.T)-{P:)-{Pa) + {P.To]-{P:)-{Pao}
where the nonlinear stiffness matrices, [Nl]s, [Nljs and [N2]ss, linearly and
quadratically depend on the thermal displacement {Ws}. The temperature
dependent nonlinear material properties are handled with small temperature
increments of AT, and the material properties are thus considered to be
constants within each small temperature increment. The initial deflection and
initial stress are both zero at the first temperature increment. The effects of
initial deflection and stress are thus included in the formulation. One effective
approach for solving Eq. (10) involves the application of Newton-Raphson
iterative method. Thus, the i-th iteration Eq. (10) can be written as
=(AP), (11)
then and {AP};^^ are updated by using
={WJ,. + {AW},,, (12)
The solution process seeks to reduce the imbalance load vector {AP}, and
consequently {AW}, to a specified small quantity (10"^ in this study). The
tangent stiffness matrix and the imbalance load vector are
+ + (13)
and
1645
{API = {P^} - {Pr'} -{PJ + [P^J- iPrl) - {P^} ,j .
-([/:]-[Ji:„^,]+[<]+[^j+[^j+MA'iL +[Afi„L +i[JV2]„,){r,},
where the nonlinear stiffness matrices [Nljsi, pSTloJsi and pSf2]ssi are evaluated
using The total thermal deflection is the sum of the converged {Ws}
from the many small temperature increments.
Random Response Analysis
The other equation derived from Eqs. (8b) and (9) is a dynamic
equation, and it can be written as
mm)},
+ (m-[^«.r] + [<] + [^<,] + [^J + [^lL +lN2l,W{,t)). (15)
= {^(0}
The sum of stiffness matrices in Eq. (15) is exactly the converged tangent
stiffness matrix from Eq. (1 1). Therefore, there is no need to assemble the
system stiffness matrices by considering the effects of SMA recovery stress,
thermal stress, and thermal deflection from each element as in the
conventional finite element approach. By set {P(t)}=0, Eq. (15) is a standard
linear eigenvalue problem, and the natural frequencies and mode shapes of
vibration about the thermally deflected position are obtained.
Substituting a modal transformation and truncation of [W], = [(|)]{g'}
into Eq. (15) with the consideration of damping, a set of uncoupled modal
equations about the thermally deflected or buckled position can be expressed
as
+ + = r=l,2,...,N (16)
where the modal mass, stiffness and force are
/. = {<!>,}’■ {7>(0} {<!>,}
Thus, the root mean square (RMS) of maximum deflections and strains can be
easily determined.
RESULTS AND DISCUSSION
The results shown in this study were based on an SMA fiber-
reinforced composite laminate, where the graphite-epoxy composite was
treated as matrix. The following material properties were used in the analysis:
SMA-Nitinol Graphite-Epoxy
T. 100°F(37.78°C) E' 22.5(155)
If 145(62.78) Ez 1.17(8.07)
1646
Gi2 0.66(4.55)
E* From Reference [2]
a * From Reference [2]
G* 3.604 Msi (24.9 GPa), T<Ts
G* 3.712 Msi (25.6 GPa), T>Ts
0.3
p 0.6067 xl0'"/6- 5^ /m\645QKg/m^)
a 5.7 X 10-^ r F(10.26 x lO"^ E Q
|Xi2 0.22
p 0.1458x10-' (1550.07)
ai -0.04xl0-®(-0.07xl0-®)
Oil 16.7x10-^(30.1x10-")
The finite element used in this investigation is the three-node Mindlin (MIN3)
plate element with improved transverse shear [22]. A reference or ambient
temperature - 10'' F and a uniform temperature distribution are used in all
the examples.
Thermal Deflection
A simply supported 15x12x0.048 in. rectangular (0/90/90/0)s laminate
with or without SMA fibers is studied in detail. The mesh size is 10x8x2 for
the full plate. Figures 1 and 2 show the maximum deflections versus
temperature for a laminate with no SMA fibers (vs=0) and with 10%, 20%,
30% SMA fibers and 3% prestrain (v^ = 10%, 20%, 30% and s, =3%),
respectively. It is seen that the thermal deflection of the panel without SMA
approximately reaches 2 times the plate thickness (Wmax/h=2) at 200'’ F . For
the panel with different volume fraction of SMA, Fig. 2 shows that the
thermal deflection drops dramatically after the SMA is activated at
transformation temperature (T>Ts) and the deflection will increase gradually
when the thermal expansion effects become dominant. The most important
phenomenon is that although the Young’s modulus of SMA is lower than that
of the composite matrix material, the thermal deflections of the panel can be
reduced because of the effects of recovery stress of SMA. It can be seen that
the thermal deflections (Wmax/h) are less than 1.0 at 200'’ jF and less than 1.5
at 300" F . Figure 3 shows the maximum deflection versus temperature for a
laminate with 30% SMA fibers and 3%, 4% and 5% prestrains. Compared to
Fig. 2, it is seen that the volume fraction of SMA fibers is more effective than
the prestrain of SMA in the reduction of thermal deflection. The thermal
deflections versus temperature for a clamped 15x12x0.048 in. rectangular
(0/90/90/0)s laminate are shown in Fig. 4. The clamped plate is more stiff than
the simply supported. The critical buckling temperature for the clamped
laminate without SMLA is slightly higher than the SMA transformation
temperature Ts. This leads to the postponement of thermal deflection until
300" F for the clamped laminate with 10% SMA and 3% prestrain. In this
clamped case, this implies that the thermal deflection can be completely
eliminated for the highest operating temperature less than 300"F . The
1647
thermal deflection in the temperature range between the critical buckling
temperature and SMA activated temperature for the simply supported case
shown in Figs. 2 and 3 can also be suppressed by: 1) selecting the proper
percentages of SMA fiber volume fraction and prestrain, and 2) altering the
transformation temperature Ts by changing the composition of alloy.
Random Response . n. , a u
For panels with SMA, the dynamic response is affected by the
components of stiffness due to SMA ([K]) and due to recovery stress of SMA
(rX’*]) Note that the Young’s modulus of SMA is lower than that of the
composite matrix, thus the panel becomes less stiff when SMA fibers are
embedded. The increase in the dynamic response, observed for some case, is
due to the relatively lower modulus and higher mass density of the SMA. On
the other hand, large inplane tensile forces are induced by the recovery stress
of SMA and this effect will decrease the dynamic response of the paneL
Figure 5 shows that the RMS(Wm»A) of the panel with 10% SMA fibers and
5% prestrain at 170°F is slightly larger than that of the panel without SMA at
nO°F. In this case, the recovery forces induced ^e not sufficient to
overcome the loss of stiffness due to the modulus deficiency of ShM fibCTS.
However, the panel with 20% and 30% SMA fibers and 5% prestrain provide
ample recovery forces to significantly reduce the p^el dynamic response.
Figure 6 shows the total maximum deflection of panel with no SMA
fibers (vs=0) and for three SMA prestrain values s, = 3, 4 and 5% for each
nonzero SMA volume fraction (vs=10, 20 and 30%) at
SSL=100dB. It clearly indicates that the six graphite/epoxy panels with SMA
volume fraction of either 20% or 30% and s, = 3, 4 and 5% are all
acceptable designs. Compared to the panel with no SMA fibers, those p^els
give much small amount of maximum RMS random deflections as well as
thermal deflections. , . r, 0+ mn
The power spectral density (PSD) of the maximum deflection at 100
dB is shown in Fig. 7 for three cases; no SMA at 10° F ; 10% SMA and 3%
prestrain at 110° F ; and 30% SMA and 5% prestrain at 1 70° F , respectively.
It is seen that the SMA fiber-reinforced plates exhibit significant peak-
amplitude reduction and frequency increase at 170° F .
CONCLUDING REMARKS
The stress-strain relations for a thin composite lamina with embedded
SMA fibers have been developed. The finite element method has been
successfully implemented to analyze the thermal deflection and random
1648
8
response of SMA fiber-reinforced composite plates with the consideration of
nonlinear material properties of SMA and nonlinearity in geometry.
With the proper percentages of SMA volume fraction and prestrain
and also the altering of transformation temperature by changing the alloy
composition, the thermal deflection can be dramatically reduced. This
reduction in thermal deflection could be useful in practical applications by
maintaining optimal aerodynamic configuration for flight vehicles and
eliminating snap-through motions.
The RMS maximum deflection can be reduced with some
combinations of SMA volume fraction and prestrain. After the SMA is
activated and the recovery forces induced are sufficient to overcome the loss
of stiffness due to the modulus deficiency of SMA, the dynamic response can
be significantly reduced.
ACKNOWLEDGMENTS
The authors would like to acknowledge the support by grant F33615-
91-C-3205, AF Wright Laboratory. Dr. Howard F. Wolfe is the technical
monitor. The authors would also like to thank Dr. Alex Tessler, NASA
Langley Research Center for his assistance on the MIN3 element.
REFERENCES
1. Tadaki, T., Otsuka, K. and Shimizu, K., Shape Memory Alloy. In Annual
Review of Materials Science, Eds. Huggins, R. A., Giordmaine, J. A.
and Wachtmank, J. B., Annual Reviews, Inc., Palo Alto, CA, 1988, Vol.
18, pp. 25-45.
2. Cross, W. B., Kariotis, A. H. and Stimler, F. J., Nitinol Characterization
Study, NASA CR-1433, 1970.
3. Baz, A. and Tampe, L., Active Control of Buckling of Flexible
'QQdms. Proceedings, 18th Biennial Conference on Failure Prevention and
Reliability, Montreal, Canada, 1989, ASMEDE-Vol. 16, pp. 211-218.
4. Chaudhry, Z. and Rogers, C. A., Bending and Shape Control of Beams
Using SMA Actuators. Journal of Intelligent Materials Systems and
Structures, 1992, Vol. 2, pp. 581-602.
5. Baz, A., Iman, K. and McCoy, L, Active Vibration Control of Flexible
Beams Using Shape Memory Actuators. Journal of Sound and Vibration,
1990, Vol. 140, pp. 437-456.
6. Segalman, D. J., Parker, G. G. and Inman, D. J., Vibration Suppression by
Modulation of Elastic Modulus Using Shape Memory Alloy. Intelligent
Structures, Materials and Vibrations, 1993, ASME-DE-Vol. 58, pp. 1-5.
7. Maclean, B. J., Patterson, G. J. and Misra, M. S., Modeling of a Shape
Memory Integrated Actuator for Vibration Control of Large Space
1649
Structures. Journal of Intelligent Materials Systems and Structures, 1991,
Vol. 2, pp. 72-94.
8. Kim, J. H. and Smith, C. R., Control of Flow-Induced Vibrations Using
Shape Memory Alloy Wires. Adaptive Structures and Material Systems,
1993, ASME AD-Vol. 35, pp. 347-34.
9. Ditman, J., Bergman, L. and Tsao, T., The Design of Extended Bandwidth
Shape Memory Alloy Actuators. Proceedings, AIAA/ASME Adaptive
Structures Forum, Hilton Head, SC, 1994, pp. 210-220.
10. Rogers, C. A., Liang, C. and Li, S., Active Damage Control of Hybrid
Material Systems Using Induced Strain Actuator. Proceedings, 32nd
AIAA/ASME/ASCE/AHS/ASC Structure, Structural Dynamics and
Materials Conference, Baltimore, MD, 1991, pp. 1190-1203.
11. Venkatesh, A., Hilbom, J., Bidaux, J. E. and Gotthardt, R., Active
Vibration Control of Flexible Linkage Mechanisms Using Shape Memory
Alloy Fiber-Reinforced Composites. Proceedings, 1st European
Conference on Smart Structures and Materials, Glasgow, England, Eds.
Culshaw, B., Gardiner, P. T. and McDonach, A., Institute of Physics
Publishing, Bristol, England, 1992, pp. 185-188.
12. Anders, W. S., Rogers, C. A. and Fuller, C. R., Vibration and low
Frequency Acoustic Analysis of Piecewise-Activated Adaptive Composite
Panels. Journal of Composite Materials, 1992, Vol. 26, pp. 103-120.
13. Liang, C., Rogers, C. A. and Fuller, C. R., Acoustic Transmission and
Radiation Analysis of Adaptive Shape Memory Alloy Reinforced
Laminated Plates. Journal of Sound and Vibration, 1991, Vol. 145, pp.
72-94.
14. White, S. R., Whitlock, M. E., Ditman, J. B. and Hebda, D. A.,
Manufacturing of Adaptive Graphite/Epoxy Structures with Embedded
Nitinol Wires. Adaptive Structures and Material Systems, 1993, ASME
AD-Vol. 35, pp.71-79.
15. Jia, J. and Rogers, C. A., Formulation of a Mechanical Model for
Composites with Embedded SMA Actuators. Proceedings, 18th Biennial
Conference on Failure Prevention and Reliability, Montreal, Canada,
1989, ASME DE-Vol. 16, pp. 203-210.
16. Rogers, C. A., Liang, C. and Jia, J., Behavior of Shape Memory Alloy
Reinforced Composite Plates, Part 1; Model Formulation and Control
Concept. Proceedings, 30th AIAA/ASME/ASCE/AHS/ASC Structure,
Structural Dynamics and Materials Conference, Mobile, AL, 1989, pp.
2011-2017.
17. Zhong, Z. W., Chen, R., R., Mei, C. and Pates, C. S., Ill, Buckling and
Postbuckling of Shape Memory Alloy Fiber-Reinforced Composite Plates.
Symposium on Buckling and Postbuckling of Composite Structures, 1994,
ASME AD-Vol. 41/PVP-Vol. 293, pp. 115-132.
1650
18. Pates, C. S., IH., Zhong, Z. W. and Mei, C., Passive Control of Random
Response of Shape Memory Alloy Fiber-Reinforced Composite Plates.
Proceedings, Fifth International Conference on Recent Advances in
Structural Dynamics, eds. Ferguson, N. S. et al. Institute of Sound
Vibration Research, Southampton, England, July 18-21, 1994, pp. 423-
436.
19. Turner, T. L., Zhong, Z. W. and Mei, C., Finite Element Analysis of the
Random Response Suppression of Composite panels at Elevated
Temperatures Using Shape Memory Alloy Fibers. Proceedings, 35^^
AIAA/ASME/ASCE/AHS/ASC Structure, Structural Dynamics and
Materials Conference, Hilton Head, SC, 1994, pp. 136-146.
20. Xue, D. Y. and Mei, C., A Study of the Application of Shape Memory
Alloy in Panel Flutter Control. Proceedings, Fifth International
Conference on Recent Advances in Structural Dynamics, eds. Ferguson, N.
S. et al. Institute of Sound Vibration Research, Southampton, England,
July 18-21, 1994, pp. 412-422.
21. Chen, R. R. and Mei, C., Thermo-Mechanical Buckling and Postbuckling
of Composite Plates Using the MIN3 Elements. Symposium on Buckling
and Postbuckling of Composite Structures, 1994, ASME AD-Vol.
41/PVP-Vol. 293, pp. 39-53.
22. Tessler, A. and Hughes, T. J. R., A Three-Node Mindlin Plate Element
with Improved Transverse Shear. Computer Methods in Applied
Mechanics and Engineering, 1985, Vol. 50, pp. 71-101.
APPENDIX
Stress-Strain Relations of a SMA Embedded Composite Lamina
A representative volume element of a SMA fiber-reinforced composite
lamina is shown in Fig. Al. The element is taken to be in the plane of the
plate. The composite matrix, for example graphite/epoxy, has the principal
material directions 1 and 2 and the SMA fiber embedded in the 1 -direction.
In order to derive the constitutive relation for the 1 -direction, it is
assumed that a stress Oj acts alone on the element (Cj =0) and that the SMA
fiber and composite matrix are strained by the same amount, Sj (i.e., plane
sections remain plane), the 1 -direction stress-strain relation of the SMA fiber
can be described as
a,, = £>,+o;. T>T, (Al)
or
Qj, = £*(8i -a.AT), T <f (A2)
where Ts is the austenite start temperature and as is the thermal expansion
coefficient. The Young’s Modulus E] and the recovery stress gI are
1651
temperature dependent, indicated by superscript (*). The recovery stress a * is
also dependent on the prestrain . For Nitinol, a* and E* can be determined
from Figs. A2 and A3, respectively [2]. Similarly, the one-dimensional stress-
strain relation in the 1 -direction for the composite matrix can be expressed as
=-E[»(Si -ottaAT) (A3)
The resultant force in the 1-direction (02 = 0) is distributed over the
SMA fiber and composite matrix and can be written as
a,.4, (A4)
where (a 1, 4^1 ), and are the (stress, cross section area) of
the entire element, SMA fiber, and composite matrix, respectively. Thus, the
average stress a, is
<^1 (A5)
where v = A / A and = A„ / A, are the volume fractions of 'SMA and
composite matrix, respectively. When T>Ts, the SMA effect is activated and
the one-dimensional stress-strain relation in the 1-direction becomes
a, = (£>, +ct;)v^ -i- £i„(s, -a,„Ar)v„
= -Ei's, +<J>. AT”
£:=£,„v„+£;v. (A7)
When T<Ts, the SMA effect is not activated and the stress o, is
a, = £,‘6, - (E'a,v^ + £,„a,„,v„, )AT
= £,*(s, -a, -AT)
where
A similar constitutive relation may be derived for the 2-direction by
assuming that the applied stress 02 acts upon both the fiber and the matrix
(a, = 0) . Thus, the one-dimensional stress-strain relations in the 2-direction
for the SMA fiber and the composite matrix become
C'2. = <^2 = -E* (82. - “.AT) (A1 0)
and
'J2«=02=£2™(s2„-a,„Ar) (All)
respectively. The recovery stress does not appear in Eq. (AlO), since the SMA
fiber prestrain s,. and recovery stress a * are considered to be a 1 -direction
effect only.
The total elongation is due to strain in the composite matrix and the
SMA fiber and may be written in the form
As. = Ajs.^ + As,, (A12)
1652
Thus, the total strain becomes
E2=S2„''„,+E2.'',
Since o, =£*(£,, -a. AT^, Eqs. (AlO) and (All) may be substituted into
Eq. (A13) to give
E'
+ (a.v,+a,„v„)Ar
Therefore, the modulus and thermal expansion coefficient in the 2-direction
become
£* - - (A15)
and
“2=a.2„i'„+a,v. (A16)
Expressions for the hybrid composite Poisson’s ratios and shear moduli follow
from similar derivations.
The constitutive relations for a thin composite lamina with embedded
SMA fibers can be derived using a similar engineering approach to give
\Q\\ Q\2
0 Ql
a,
0
0 ^
0
v. J
y^AT, T>T, (A17)
a;
0 '
a;'
\
Q22
0
<
:S2
> _ <
>AT
0
av
v
[Vl2,
0 ^
/
= O; O; 0 s, - a,Ar, T<T, (A18)
^ ^ See.V.Yuj >
where [0]^ and [Q*] are the reduced stiffness matrices of the composite
matrix and the composite lamina, respectively. The [Q*] matrix is temperature
dependent and is evaluated using the previously derived relations as
(■E*. Ibz) = (£.». i^.2..)‘'« + (C. 11.)'',
and
(E' G') = - - . (A20)
where the |i’s are Poisson’s ratios and the G’s are the shear moduli. The
thermal expansion coefficients a* and a2are derived in Eqs. (A9) and (A16).
1653
1654
Vi% 0 10 20 30
Figure 6, Total Maximum Deflection/Thickness at HO^F and 100 dB SSL.
Frequency (Hz) element for SMA flber-relnforced
Figure?. Power Spectral Density of Wmax. hybrid composite lamina.
Temperature, “F Temperature,
Figure A2 - SMA recovery stress versus temperaturepigure A3 - SMA modulus versus temperature
and initial strain (Cross et al.,1970). (Cross et al., 1970).
1655
1656
SIGNAL PROCESSING!
DESCRIPTION OF NON-LINEAR
CONSERVATIVE SDOF SYSTEMS
Michael Feldman Simon Braun
Faculty of Mechanical Engineering
Technion — Israel Institute of Technology
Haifa, 32000, Israel
1 INTRODUCTION
There are several methods developed for the analysis and identification of non-linear dy¬
namic systems and vibration signals [2, 8, 7, 9]. In the recent past the characterization
of the response of non-linear vibration systems has been approached using the Hilbert
transform in time domain [4, 3]. The objective was to propose a methodology to identify
and classify various types of non-linearity from measured response data. The proposed
methodology concentrates on the Hilbert transform signal processing techniques essentially
on signal envelope and instantaneous frequency extracting, which enables us to estimate
both systems instantaneous dynamic parameters and also elastic (restoring) and friction
(damping) force characteristics. When the traditional phase plane (t/,y) of a non-hnear
system is replaced by the complex plane (y,y), the signal envelope, the instantaneous
frequency, and the obtained backbone takes an unusual fast osciEation (modulation) form
[1]. In addition, the obtained envelope of the signal gains a bias in comparison with the
phase plane ampHtude [1]. This naturaUy occurring fast modulation and the bias requires
more detail investigation. Further use of the Hilbert methodology for non-linear structures
also invites more sophisticated analysis.
2 PHASE PLANE REPRESENTATION
The differential equation of motion for SDOF non-linear conservative vibration system
may be written as my K{y) = 0, where m - the mass, y - the acceleration, and K{y)
- the restoring force which is a function of displacement y. The restoring force contains
the linear stiffness and also any additional non-linear restoring force component. The
second-order differential equation of a conservative system will then take the general form
y + A:(i/) = 0 (1)
1657
Figure 1: Phase plane (dash Hne) and Analytic signal (bold line) representation of Duffing
equation (e = 5)
where the term k{y) - K{y)lm represents the restoring force per unit mass as a function
of the displacement y.
Traditionally we introduce a new variable y, which enables us to exclude time from
the equation of motion although y and y are still time dependent, so y = ^ = ^y. In the
new coordinates Eq.(l) takes the Mowing form: f = ^- Using of the new variable y is
a traditional way of study the motion of an oscillator by representing this motion on the
y, y plane (Fig. 1), where y and y are orthogonal cartesian coordinates [2]. Hence, the
time for a full cycle becomes
where ymax - the maximum value of the displacement, so the velocity corresponding to
ymax hi an extreme position is zero, /(*) = fc(*). It is a well known fact that the expression
obtained for T is a function of y^ax, thus for non-linear systems the period of oscillation
depends on the total stored energy (non-isochronism of oscillation in non-linear system).
We shall now divide the angle corresponding to the quarter of a complete circle into n
equal small sectors dcf) = Next, we use Eq.(2) to estimate the integral, but separately
for each i small interval with new limits; from // = co5[| — d(f>{i — 1)] to — cos(| — i)
where u = y fym&x- In- this case only for a linear restoring force the last equation results in
a constant value 27r. In general case of non- linear system the current period is a varying
function of a phase angle {Ti{<l>) i=^ const). An inverse function of the current period will be
called an curvtnt angular frequency uJi{(l>) = The current angular frequency does not
remain constant, but fluctuates between a maximum and a minimum as the radius vector
rotates between the y - and y-axes of the phase plane. Also the radius vector of phase
plane r{4>) becomes to modulate, from its minimum up to the maximum value r{(j>) =
1658
bm«. [•!
Figure 2: Solution (a) and periods (b) of the Duffing equation (e = 5, n — 100, bold line
- the current period, dash line - the mean value of period
+ y2(<^) = ^y(<^y -h 2 To illustrate this interesting phenomena of the
current period modulation let us consider an example of Duffing equation.
y {I + ey^)y - 0 (^)
where e is the non-linear parameter. For the purpose of calculating the current pe¬
riod of free vibration, we substitute expression Eq.(4) into Eq.(3) and obtain T{i) =
4 Figure 2, shows the free vibration signal obtained via numeric
V{i-«D(i+'(i+«D7^' . . , , , w j j
integration of the Duffing equation together with the calculated current period ol the so¬
lution for the total number of small sectors n = 100. It is clear that the the current
period oscillates two times faster than the non-linear solutions. Now it is clear that the
known direct integration for the total period Eq.(2) produces only an average value T
which corresporids to the total cycle of the motion.
3 COMPLEX PLANE REPRESENTATION OF AN¬
ALYTIC SIGNAL
Let us consider another important technique of representation of vibration process in the
time domain on the base of rotating vectors. According the analytic signal theory real
vibration process t/(t), measured by, say, a transducer, is only one of possible projections
(the real part) of some analytic signal F(£). Then the second projection of the same
signal (the imaginary part) y{i) wiU be conjugated according to the Hilbert transform.
Using the traditional representation of analytic signal y{t) in its complex, trigonometric
or exponential form Y{t) = y{t) + iy(t) = iy(0l [ cos m + j sin ^(t) ] = A{t) one
can determine its instantaneous amplitude ( envelope, magnitude)
A(t) = \Y{t)\ = y/ym+m = (5)
and its instantaneous phase
V,(f) = arctan ^ = Im[liiy(t)l (6)
1659
The instantaneous angular frequency is the time- derivative of the instantaneous phase:
u;{t) = -0(f) =
yWyji) - y(0y(0 _
Am
= Im
Y{t)
noj
(7)
Each point on complex plane is characterized with the radius vector -A(f), and the polar
angle 0 (Fig. 1). The analytic signal method is an expedient effective enough to solve gen¬
eral problems of vibration theory, among them - analysis of free and forced nonstationary
vibrations (transient processes) and non-linear vibrations.
3.1 The Hilbert transform
The single-value extraction (demodulation) of an envelope and other instantaneous func¬
tions of a signal is an issue based on the Hilbert integral transform [7]. The Hilbert Trans¬
form of a real- valued function a;(f) extending from — co to -j-oo is a real- valued function
defined by H[y(t)] = y(f) = J dr, where 3/(f) - the Hilbert transform of the initial
process y{t), and meaning of the integral implies its Cauchy principal value. Thus y{t) is
the convolution integral of y{t) with (l/7rf), written as y(f) = y{t) * (l/-7rf) . The double
Hilbert transform yields the original function with an opposite sign, that’s it carries out
shifting of the initial signal in — x. The power (or energy) of a signal and its Hilbert trans¬
form are equal. For n[t) lowpass and y{t) highpass signals with nonoverlapping spectra
[7],
H[n(f)2/(f)] = 7i(f) y(f). (8)
3.2 Approximate non-linear system representation
The showning non-linear restoring force k{y) of the second-order conservative system Eq. 1,
can be recast into multiplication of varying non-linear natural frequency a;o(3/) and of the
system solution y+ojl{y)y - 0. Let’s assume that the varying non-linear natural frequency
uo{y) could be separated into two different parts [5]. The first part o^o is much slower and
the second component ct>i(y) is faster than the system solution, so the equation of motion
will be _
y-\-[uj^+ ujl{y) ] y = 0 (9)
The proof of the decomposition of a signal into a sum of low- and high-pass terms, based
on Bedrosian’s theorem about the Hilbert transform of a product, could be found in [6].
Now according to nonoverlapping property of the Hilbert transform Eq. (8 ) we use the
Hilbert transform for both sides of Eq. 9 y -f u>Qy -h (^i{y)y = 0. Multiplying each side of
the last equation by j and adding it to the corresponded^ides of Eq. 9 we get a differential
equation in the Analytic signal form Y 4- (jJqY + [ + ywj ]y = 0, where Y = y~\- jy. This
complex equation can be transformed to the commonly accepted form
y + j5oir + a;o\F = 0 (10)
where Wqj = wJ -h - the varying natural frequency, <^oi = - the fast
varying fictitious friction parameter, and Y = y jy - is the complex solution.
1660
The obtained equation of the non-linear system has the varying natural frequency,
consisting of slow ZUq and fast component, and also has the fast varying fictitious friction
parameter ^oi- It should be pointed that this non-stationary equation is not a real equation
of motion. It is just an artificial fictitious equation which produce the same non-linear
vibration signal.
3.3 Non-stationary equation of motion in the analytic signal
form
Taking into account the analytic signal representations, enables one to consider this equiv¬
alent equation of motion so as to estimate instantaneous natural frequency and instanta¬
neous damping coefficient.
Some general form of a differential equation of motion in the analytic signal form could
be written
Y -f- 2ho{A)Y -f u;l{A)Y = 0 (U)
where Y - the system solution in the analytic signal form, ho - the instantaneous damping
coefficient, Wq - the instantaneous undamped natural frequency. This equation of motion
will have varying coefficients that satisfied the envelope and instantaneous frequency of
non-linear oscillated solution.
Solving two equations for real and imaginary parts of Eq.(ll), we can write the expression
for non-stationary coefficients as functions of a first and a second derivative of the signal
envelope and the instantaneous frequency.
"oW=“ - 1 + + a;
, ^ A Cj
Hi) = -7 -
(12)
where ufo{t) - instantaneous undamped natural frequency of the system, ho{t) - instan¬
taneous damping coefficient of the system, to, A - instantaneous frequency and envelope
(amplitude) of the vibration with their first and second derivatives (cl;, A, A).
3.4 Two-component signal representation
Let us consider a conservative system having a single degree of freedom that are gov¬
erned by simple non-linear differential equation having the form Eq. 1 Assuming k can be
expanded, one can rewrite Eq. 1 as [8]
n-l
where an = and denotes the nth derivative with respect to the argument.
We assume that the solution of Eq. 13 can be represented by an expansion having the
form y{t, e) = £yi(t) + £^y2(t} + £%(t) + - . • where the y^ are independent of e. Retaining
1661
Table 1: Formulas for extreme and mean values of instantaneous frequency and amplitude
of bibarmonics
Inst, amplitude
Inst, frequency
•^min “ -^1 •^2
Am&x = Al A A2
A = Ai^l + Al/Al
two first main terms of the solution we will get an equation where each of amplitude and
frequency is a function of linear and non-linear parameters of the system (Eq. 13).
y(t'^ = + 2/2 = -^1 cos Wit -f A2 cos W2t (14)
Universally known that the more close the frequency of the first term to the precise solution
27z JT (Eq. 2) the better approximation.
Consider the non-linear solution that consists of two quasi-harmonics each one with
different amplitude and frequency in time domain. In this case, the signal can be modeled
as a weighted sum of two monocomponent signals, each one with its own instantaneous
frequency and amplitude function. Assuming that each individual signal has a large band¬
width and time duration product (with a purely positive IF), application of the Hilbert
transform for both sides of Eq.(14) will produce an analytic signal of the form.
r(t) = + (15)
The envelope and the instantaneous frequency of double- component vibration signal
Y{t) are:
A{t) = {Al +Al-\- 2A1A2 cos[(a;2 - a;^)^]}^/^
a;(t) = ctJi -i-
Al -i- A1A2 cos[(a;2 - ^i)t]
(w2-c,i)-^A^it)
(16)
(17)
From Eq.(l6) it can be seen that signal envelope consists of two diiferent parts, that
is a constant part included sum of component amplitudes squared Al, Al and also a
fast varying (oscillating) part, the multiplication of these amplitudes with function cos of
relative phase angle between two components. The same inference could be made about
the instantaneous frequency of the two-terms solution. The obtained formulas for extreme
and mean values of instantaneous frequency and amplitude of the biharmonics are shown
in Table 1.
It is significant that the mean value of the instantaneous amplitude is very close to the
corresponded amplitude of the first harmonics (A Ai). Also mean value of the instanta¬
neous frequency agrees closely with the frequency of the first harmonics {w ^ wi). Again
consider an example of the Duffing equation (Eq. 4). Using for instance the method of the
analytic signal representation [9] one can get an equation for the first order approximation
of the free vibration frequency
a.? = 1 + ^ (18)
1662
Angulw fr*qu«ncy
Figure 3: Backbone of the Duffing system (s = 5)
b
Figure 4: The solution (a) and the instantaneous frequency (b) of the Duffing equation
(e = 5)
where a — j/max is some maximum of the displacement. This approximation of free vi¬
bration frequency (dot line on Fig. 3) maps an approximate backbone very close to the
traditional precise backbone (bold line on Fig. 3).
Let now an approximate solution be in a form which includes the first two harmonics only
y[t) = yi{t) + y2{t) - Aicosujit -f A2 cos aj2t (19)
where 0^2 = 3a;i, A2 = Ai - I - A2 , K - numerical coefficient. The instantaneous
frequency and the envelope of the solution derived from the Hilbert transform takes the
form of oscillated functions with double frequency (Fig. 4). The approximate double
component solution {e = 5, K = 156) coincides very closely with the numerical solution
(Fig. 4, a). The corresponded envelope (dash-dot line), and instantaneous frequency (b)
of the Duffing equation is also plotted in Figure 4.
Maximum values of the envelope A{t) (Eq. 16) are equal to the chosen maximum of
displacement a, but all other points of the envelope (dash-dot hne on Fig. 4, a) including
the average value of amplitude A are smaller than the maximum of displacement a. Now
it is clear that for non-hnear conservative vibrations two different estimations of amplitude
could be suggested. The first estimation is the maximum of displacement which is
1663
equal to the maxiiiium of envelope y^aax ~ ^max- The second estimation is the average value
of envelope A. The difference between the average value of envelope A and the maximum
of displacement a could be considered as a small bias of amplitude of the solution.
It is vital to note that the mean value of the instantaneous frequency Q obtained
through the Hilbert transform is equal to the theoretical value of the free vibration fre¬
quency (Eq. 2 ). On Fig. 4, b the dash line shows these two coincident functions. Q = 2Tr/T
This fact indicates that the Hilbert transform is best representation of non-linear vibra¬
tions,
3.5 Free vibration frequency and amplitude dependence
Because we considered several different representations of vibration amplitude it is evident
that we get some different backbone depictions. A traditional theoretical precise backbone
of non-linear system is a dependency between the average free vibration frequency a> (cor¬
responded to the average total cycle of vibration, Eq. 2) and the maximum displacement
Vmax (Fig. 3, bold line). For a conservative system with a chosen maximum displacement,
the theoretical backbone is no more than a point (-}- on Fig. 3). As far as we can use the
mean value of the envelope A, a dependency between this mean value and the average
free vibration frequency cD could be a new kind of the theoretical backbone (dotted line on
Fig. 3). But the difference between these two backbone definitions is very small and cor¬
responds to the small bias. According to the traditional vibration system representation,
conservative non-linear system has a constant amplitude. It means that vibrations of a
conservative Duffing system should map on backbone plot just as a point with a constant
amplitude and frequency.
A somewhat different backbone for a conservative system is gained according to the in¬
stantaneous frequency and amplitude representation. Eliminating the time dependent os¬
cillating part cos[{uj2-^i)t] from Eq.(16) and Eq.(17), we shall find the equation between
signal instantaneous characteristics (envelope and frequency) and initial four parameters
of signal components
(20)
uj\ — 2c4;
Equation (20) determines the signal envelope A as a function of instantaneous frequency
ui in the form of an hyperbola, whose length and curvature depends on four initial pa¬
rameters of the biharmonics. So according to the Hilbert transform the instantaneous
backbone of a non-linear conservative system stretches to a very short hyperbola (dashline
on Fig. 3). This short length of hyperbola is the same for both the numerical solution and
the approximate double component solution. The hyperbola goes very close both to the
traditional theoretical point of maximum displacement (+) and to the approximate mean
value of envelope (Fig. 3, o). Figure 3 also includes two additional backbones (short dash
lines) corresponding to different maximum values (a) of the displacement.
3.6 Real non-linear force estimation
The proposed direct time domain method based on the Hilbert transform allows a direct
extraction of the linear and non-linear parameters of the system y -f uJoivYy = 0 Horn the
1664
Displacvmvnt
Figure 5: Cubic force characteristics of Duifing model (e = 5)
measured time signal y of output. The resulting non-linear algebraic equations (Eq.l2)
are rather simple and do not depend on the type of non-hnearity that exists in the struc¬
ture. When applying this direct method for transient vibration, the instantaneous modal
parameters are estimated directly, with the corresponding equation
y + ^oiV + ^oiV ~ ^
where ioh - is the fast varying natural frequency, 5oi “ is the fast varying fictitious friction
parameter, y - is the displacement, and y - is the Hilbert transform of the displacement.
With this representation of a non-linear solution we can try to solve an inverse identifi¬
cation problem, namely, the problem of estimation of the initial non-linear elastic force
characteristics.
3.6.1 Decomposition technique
Let us consider the case of a conservative system with an initial non-linear spring. Accord¬
ing to Eq. 11 this real non-linear elastic force wiU produce two different fictitious members
(elastic and hysteretic damping). The new restoring force, here assumed to be a function
of displacement y and its Hilbert transform y. This restoring force includes both the fast
hysteretic damping 6oiy and the fast elastic force u^l-^^y. It means, that the initial non¬
linear spring force <jJo{yyy is split into two terms, and, by summing over the terms, the
initial non-linear force characteristics can be extended. Therefore a simple composition of
these two members of equation of motion (Eq. 21) will result in the real non-linear force
characteristics:
k[y{t)] = 2ho{t)y-^L,l{t)y (22)
where k[(y)] is the real elastic instantaneous force, /io(0 instantaneous damping
coefficient, Wo(0 is the instantaneous undamped natural frequency. Fig. 5 shows an exam¬
ple of the instantaneous elastic force identification for the Duffing system (Eq. 4). Fig. 5
includes the results of the identification according to formula Eq. 22 together with the
initial cubic force characteristics k{y) = (1 + 5y^)y, but these lines agree so closely that
^Kere is no a difference.
1665
3.6.2 Scaling technique
Instead of the previous identification of the detail force characteristics, we can express only
the relation between the maximum of elastic force and the maximum of displacement. It
can be determined by following consideration. The total energy of a conservative vibration
system is constant. During free vibration of the corresponded fictitious model (Eq. 11), for
each moment the energy is partly kinetic, partly potential, and partly fictitious alternating
positive or negative damping.
To estimate maximum of elastic force we can find time points when all energy is stored
in the form of strain energy in elastic deformation and the fictitious damping energy is
zero. Using for instance the biharmonics representation of non-linear vibrations (Eq.l5)
one can show that these time points correspond to the maximum of displacement. This
is an important conservative vibration system property, that around every peak point of
the displacement the corresponding value of the velocity is equal to zero and vice versa.
Therefore around every peak point of the displacement the contribution of the velocity in
the varying instantaneous elastic force is negligibly small: y{ti) = A(ti), y{ti) = 0. The
number of these peak points is far less than a total number of points of a vibration signal.
Therefore it could be recommended to extract the envelope of the fictitious elastic force
ufQ{t)y to obtain the average value of the envelope. The obtained average envelope has
a small bias relative to maximums of the spring force. Using the Hilbert transform and
the obtained expressions we can extract the maximum values of non-linear elastic force
corresponded to the maximum of displacement (circle and star point on Fig. 5).
4 CONCLUSIONS
We can draw the following conclusions from the analytic signal representation.
Whatever the method of non-linear vibration representation, both the instantaneous fre¬
quency and the amplitude of free vibration is a complicated signal. Non-linear solution
could be represented by an expansion of members with different frequencies or by a time
varying signal with oscillated instantaneous frequency and envelope.
The instantaneous frequency and envelope of non-linear vibration obtained via the Hilbert
transform are time varying fast oscillating functions. For example in the presence of a cubic
non-linearity and a threefold high harmonics, the frequency of the instantaneous parameter
oscillation is twice that the main frequency of vibration.
The mean value of the instantaneous frequency obtained through the Hilbert transform
is equal to the theoretical total cycle average value of the free vibration frequency. The
maximum value of vibration envelope is equal to the maximum of displacement. These
facts indicates that the Hilbert transform is one of the best representations of non-linear
vibrations. Naturally there is a small difference (the bias) between the average value of
the envelope and its maximum value.
The dependency between the average envelope and the average instantaneous frequency
plots the backbone that practically coincides with the theoretical backbone of non-linear
vibrations.
Using the proposed Hilbert transform analysis in time domain we can extract both the
instantaneous undamped frequency and also the real non-linear elastic force characteristics.
1666
References
[1] P. Adamopoulos, W. Fong and J.K. Hammond. Envelope and instantaneous phase
characterisation of nonlinear system response. Proc. of the VI Int. Modal Analysis
Conf., 1988, pp. 1365 - 1371
[2] A.A. Andronov, A. A. Vitt, and S.E. Khaikin. Theory of oscillators. Pergamon Press,
1966, 815 p.
[3] M. Feldman. Non-linear system vibration analysis using Hilbert Transform — I: Free
vibration analysis method ‘FREEVIB’, Mechanical Systems and Signal Processing,
1994, 8(2), pp. 119 - 127.
[4] M. Feldman, S. Braun. Analysis of typical non-Hnear vibration systems by using the
Hilbert transform. Proc. of the XI Int. Modal Analysis Conf., Kissimmee, Florida,
1993, pp. 799 - 805.
[5] M. Feldman, S. Braun. Non-linear spring and damping forces estimation during free
vibration. Proc. of the ASME Fifteenth Biennial Conference on Mechanical Vibration
and Noise Conf., Boston, Massachusetts, 1995, V 3, 1241-1248 pp.
[6] Stefan L. Hahn. The Hilbert Transform of the Product a{t)cos{ujot -j- 4>o)- Bulletin of
the Polish Academy of Sciences, Technical Sciences, Vol. 44, No 1, 1996, pp. 75-80.
[7] Sanjit K. Mitra and James F. Kaiser. Handbook for digital signal processing. Wiley-
Interscience, 1993, 1268 p.
[8] AH H. Nayfeh. NonHnear oscillations. Wiley-Interscience, 1979, 704 p.
[9] L. Vainshtein, D. Vakman. Frequencies separation in the theory of vibration and waves
(in Russian). 1983, Moscow, Nauka, 228 p.
1667
1668
A RATIONAL POLYNOMIAL TECHNIQUE FOR
CALCULATING HILBERT TRANSFORMS
N.E.King
Ford Motor Company Ltd.
Research and Engineering Centre
Laindon
Basildon
Essex SS15 6EE
United Kingdom
K.Worden
Dynamics Research Group
Department of Mechanical Engineering
University of Sheffield
Mappin Street
Sheffield SI 3JD
United Kingdom
Abstract
This paper presents a new technique for calculating Hilbert trans¬
forms of nonlinear system Frequency Response Functions (FRFs).
The method employs rational function and pole-zero decomposi¬
tions of the FRF. The new method allows Hilbert transforms for
zoomed, or generally truncated, data without the use of correction
terms. The method is validated using a computer simulation of a
Single Degree- Of- Freedom (SDOF) nonlinear oscillator.
INTRODUCTION
It is well-known that the occurrence of nonlinearities in engineering
structures can have a significant effect on their behaviour. The most spec¬
tacular examples of this can be found in the literature relating to chaotic
systems; the response of such a system to a deterministic excitation can
be unpredictable beyond a short time scale. Also, any stability analysis
for a nonlinear system will depend critically on the type of nonlinearity
present. The question of detecting structural nonlinearity is therefore of
some importance.
The Hilbert transform is now established as a means of diagnosing
structural nonlinearity on the basis of measured Frequency Response Func¬
tion (FRF) data [8]. It is essentially a mapping on the FRF G{to),
n[Gi.)] = Giu)^-yy§^ (1)
(where the integral is to be understood as a principal value). This mapping
reduces to the identity on those functions corresponding to linear systems.
For nonlinear systems, the Hilbert transform results in a distorted version
G, of the original FRF, with the form of the distortion often yielding some
indication of the type of nonlinearity.
1669
The origin of the distortion is well-known [10]; suppose is decom¬
posed so,
G{u>) = + G {u) (2)
where G^{uj) (resp. G~{uj)) has poles only in the upper (resp. lower) half
of the complex w-plane. It is a straightforward exercise in the calculus of
residues [9], to show that (with a; on the real line),
_ 1 r ^ _ = _i_ (3)
— iJj — p
if p is in the upper half-plane, and,
i7:J~Qo{Q—p){Q> — uj) uj — p
if p is in the lower-half plane. It follows that,
nlG^iuj)] = ±G^{u;) (5)
The distortion suffered in passing from the FRF to the Hilbert trans¬
form will then be given by the simple relation,
AGiuj) = n[G{uj)] - G{u;) = -2G‘{iv) (6)
This relation has already yielded some interesting results [4] and is the
basis of the technique presented in this paper. The advantages of the new
approach over the standard time- or frequency- domain methods will now
be explained.
One of the major problems in using the Hilbert transform on FRF data,
occurs when non-baseband (data which does not start at zero frequency) or
band-limited data is employed. Practically speaking, all data will fall into
one of these categories and the problem of neglecting the contribution of
the ‘out of band’ data always exists. Establishing baseband data is rarely
a problem in simulation work, but can be a problem with experimental
testing. For example, it is known [5], that the use of random excitation
for nonlinear systems produces a FRF which is invariant under the Hilbert
transform. This means that other excitation signals must be used. Most
commonly, a stepped-sine signal is used, and this is time-consuming due to
the need to reach a steady-state condition at each frequency increment. As
a consequence, this excitation is usually applied over a limited frequency
range, say to
Unfortunately, the idealised Hilbert transform of (1), is an integral over
a doubly-infinite frequency range. In practice, the data over the intervals
(-00, (jJrrvin) and oo) will be unavailable- By making use of the parity
1670
(oddness or eveness) of the real and imaginary parts of the FRF , the Hilbert
transform can be cast in the following form,
^G{uj)
9 roo
- dO.
TrJo
QGiQ) n
n-a;2
(7)
2uj r
TT Jo
dQ,
(8)
and now the integrals are over the range 0 to oo; data will be missing from
the intervals (0,a;^in) and {u^max, oo).
The problem of truncated FRF data is usually overcome by adding
correction terms to the Hilbert transform evaluated from uimin fo u^max-
general, two corrections are needed, one for each of the missing intervals.
In order to display the inconvenience of the procedure, a review of the
relevant theory follows. It will later be shown that the method proposed in
this paper circumvents the problem of the missing intervals i.e. truncation
of the data.
CORRECTION TERMS
There are essentially three methods of correcting Hilbert transforms,
they will be described below in order of complexity.
The Fei Correction Method
This approach was developed by Fei [2] for baseband data and is based
on the asymptotic behaviour of the FRFs of linear systems. The form of
the correction term is entirely dependent on the FRF type; receptance, mo¬
bility, or inertance. As each of the correction terms is similar in principle,
only the term for mobility will be described.
The general form of the mobility function for a linear system with
proportional damping is,
G(‘-) = E ^
where: Ak is the complex modal amplitude of the mode; u>k is the
undamped natural frequency of the mode and (k is its viscous damping
ratio. By assuming that the damping is small and that the truncation
frequency, tOmax is much higher than the natural frequency of the highest
mode, equation (9) can be reduced to (for uj > tUmax),
= (10)
1671
which is an approximation to the ‘out of band’ FRF. This term is purely
imaginary and thus provides a correction for the real part of the Hilbert
transform via equation (7). No correction term is applied to the imaginary
part as the error is assumed to be small under the specified conditions.
The actual correction is the integral in equation (1) over the interval
oo). Hence the correction term, denoted C^{uj), for the real part of
the Hilbert transform is,
oo
dO.
fi5(G(n)) 2 “ dCl
which, after a little algebra [2], leads to,
TTU \idJnax — w
(11)
(12)
The Haoui Correction Method
The second correction term which again, caters specifically for baseband
data, is based on a different approach. The term was developed by Haoui
[3], and unlike the Fei correction has a simple expression independent of
the type of FRF data used. The correction for the real part of the Hilbert
transform is,
Wmas
dQ
a^G{n))
n2_a;2
(13)
The analysis proceeds by assuming a Taylor expansion for G{u;) about
and expanding the term (1 — using the binomial theorem.
If it is assumed that cj^ax is not close to a resonance so that the slope
dG{co)/duj (and higher derivatives) can be neglected, a straightforward cal¬
culation yields.
C^{u;) = C^{0) -
^{G{co„
.))
■ +
+ ...
(14)
where G^(0) is estimated from,
G^(0) = 3J(G(0)) -
"Ulrna.s
j da
0+e
S(g(ll))
a
(15)
Using the same approach, the correction term for the imaginary part,
denoted by Can be obtained,
1672
TT
5
O) UJ U}
- h ^ - H
3u^3 ' 5t^5
’-'“'max “ max
+ ...
(16)
The Simon Correction Method
This method of correction was proposed by Simon [7]; it allows for trun¬
cation at a low frequency, Umiu, and a high frequency It is therefore
suitable for use with zoomed data. This facility makes the method the most
versatile of the three. As before, it is based on the behaviour of the linear
FRF, say equation (9) for mobility data. Splitting the Hilbert transform
over three frequency ranges; and (w^arjOo), the
truncation errors on the real part of the Hilbert transform, at low
frequency and the now familiar at high frequency, can be written
as,
B-
V) = -f/
2 T”,„fi&(G(w))
dO-
(17)
and
= /
do.
as{G{u))
f22_u.2
(18)
If the damping can be assumed to be small, then rewriting equations
(17) and (19) using the mobility form (9), yields.
Wmtn Jif
(19)
and
Evaluating these integrals gives,
(20)
N
( (^max + UJk){}^k -
\(^max ^fc)(^/s d" ^min^ )
\ (^ ^mtn/(^max j
The values of the modal parameters Ajt and Uk
initial modal analysis.
(21)
are obtained from an
1673
Summary
None of the three correction methods can claim to be faultless; trunca¬
tion near to a resonance will always give poor results. Considerable care
is needed to obtain satisfactory results. The Fei and Haoui techniques
are only suitable for use with baseband data and the Simon correction re¬
quires a prior modal analysis. The following section proposes an approach
to the Hilbert transform which does not require correction terms and thus
overcomes these problems.
THE RATIONAL POLYNOMIAL METHOD
The basis of this approach to calculating a Hilbert transform is to
establish the position of the FRF poles in the complex plane and thus
form the decomposition (2). This is achieved by formulating a Rational
Polynomial (RP) model of the FRF over the chosen frequency range and
then converting this into the required form via a pole-zero decomposition.
A general FRF may be expressed as the sum of a number of modes,
e.g. equation (9) for the mobility form. This ’summation’ form is readily
expanded into a rational polynomial representation,
GM = Ig (22)
where.
TIQ Tip
iz=0 i=0
and the polynomial coefficients and &», are functions of the natural fre¬
quencies, dampings and Ak of the modes. Modal analysis methods in the
frequency-domain assume either a summation form (9) or an RP form (22).
The coefficients are determined using optimisation schemes which find the
parameters which best fit the data. Various methods exist which max¬
imise the numerical conditioning and precision of the resulting coefficients.
The RP models fitted for the current work use an ‘instrumental variable’
approach which minimises the effects of noise and produces unbiased coef¬
ficients. The details of the method can be found in [1].
Once the RP model Grp is established, it can be converted into a
pole-zero form,
n
GrpH = - (24)
-Pi)
1674
where qi and pi are the (complex) zeroes and poles respectively of the
function. The next stage is a long division and partial-fraction analysis in
order to produce the decomposition (2). If p^ (resp. p~) are the poles in
the upper (resp. lower) half-plane, then,
N+
G'^p{u)) = ^
Gpp(u}) =
cr
(25)
where and C~ are coefficients fixed by the partial fraction analysis, N+
(resp. iV_) is the number of poles in the upper (resp. lower) half-plane.
Once this decomposition is established, the Hilbert transform follows from
(6). (It is assumed here that the RP model has more poles than zeros.
If this is not the case, the decomposition (2) is supplemented by a term
G°{uj) which is analytic. Equation (6) which is used to compute the Hilbert
transform is unchanged.)
The procedure described above is demonstrated in the following section
using data from numerical simulation.
SIMULATION DATA
A simulated Duffing oscillator system with equation of motion,
y + 20y + lOOOOy -f 5 x 10^/ = X (26)
was chosen to test the calculation technique. Data was generated over 256
spectral lines from 0 to 38.4 Hz in a simulated stepped-sine test based on
a standard fourth-order Runge-Kutta scheme [6]. The data was truncated
by removing data above and below the resonance leaving 151 spectral lines
in the range 9.25-32.95 Hz.
Two simulations were carried out. In the first, the Duffing oscillator
was excited with X = 1.0 N giving a change in the resonant frequency
from the linear conditions of 15.9 Hz to 16.35 Hz and in amplitude from
503.24x10“® m/N to 483.0x10"®, m/N. The FRF Bode plot is shown
in Figure 1, the cursor lines indicate the range of the FRF which was
used. The second simulation took X = 2.5 N which was a high enough
to produce a jump bifurcation in the FRF. In this case the maximum
amplitude of 401.26x10"® m/N occurred at a frequency of 19.75 Hz. Note
that in the case of this nonlinear system the term ‘resonance’ is being used
to indicate the position of maximum gain in the FRF.
RESULTS
The first stage in the calculation process is to establish the RP model
of the FRF data. On the first data set with AT = 1, in order to obtain an
accurate model of the FRF, 24 denominator terms and 25 numerator terms
1675
were used. The number of terms in the polynomial required to provide an
accurate model of the FRF will depend on several factors including: the
number of modes in the frequency range, the level of distortion in the data
and the amount of noise present. The accuracy of the RP model is evident
from Figure 2 which shows a Nyquist plot of the original FRF, G[(jj) with
the model Grp{uj) overlaid on the frequency range 10 - 30 Hz.
The next stage in the calculation is to obtain the pole-zero decompo¬
sition (24). This was accomplished here by solving the numerator and
denominator polynomials using a computer algebra package. The zeros
and poles are given in Tables 1 and 2 respectively. It may not be nec¬
essary to obtain the zeroes of the model, this is under investigation. In
order to check the stability of the root extraction, the pole-zero model was
compared with the measured data, an exact overlay was obtained.
The penultimate stage of the procedure is to establish the decompo¬
sition (2). Given the pole-zero form of the model, the individual pole
contributions are obtained by carrying out a partial fraction decomposi¬
tion, because of the complexity of the model, a computer algebra was again
used.
Finally, the Hilbert transform is obtained by flipping the sign of
the sum of the pole terms in the lower half-plane. The result of ths calcu¬
lation for the low excitation data is shown in Figure 3 in a Bode amplitude
format. The overlay of the original FRF data and the Hilbert transform
calculated by the RP method is given, the frequency range has been limited
to 10 - 30 Hz.
A simple test of the accuracy of the RP Hilbert transform was carried
out. A Hilbert transform of the low excitation data was calculated using
a standard FFT based technique [5] on an FRF using a range of 0-50 Hz
in order to minimise truncation errors in the calculation. Figure 4 shows
an overlay of the RP Hilbert transform (from the truncated data) with
that calculated from the FFT technique. The Nyquist format is used. The
figure shows that a satisfactory level of agreement has been achieved with
the new calculation method.
The second, high excitation, FRF used to validate the approach con¬
tained a bifurcation or ‘jump’ and thus offered a more stringent test of the
RP curve-fitter. A greater number of terms in the RP model were required
to match the FRF. Figure 5 shows the overlay achieved using 32 terms in
the denominator and 33 terms in the numerator. There is no discernable
difference. Following the same calculation process as above leads to the
Hilbert transform shown in Figure 6, shown with the FRF.
CONCLUSIONS
A method of calculating Hilbert transforms from truncated Frequency
Response Function (FRF) data is presented which has no need for cor-
1676
rection terms. The rational polynomial (RP) technique operates without
needing the functional form of the FRF data, it applies equally well to
receptance, mobility and inertance forms.
The main limitation in the method is in the ability of the RP curve-fitter
to match the measured FRF. Sensitivity of the method to noise must be
investigated. A further problem relates to uniqueness and overfitting of the
model. This has already been addressed in the context of modal analysis
e.g. through the use of stability plots, and the solutions discovered there
should apply.
The implementation of the algorithm used here was based on C code
and computer algebra. It is intended to produce a single unified program
to carry out the various steps.
ACKNOWLEDGEMENTS
The authors would like to thank Dr. Pete Emmett for providing the
rational polynomial curve-fits used in this work.
References
[1] Emmett (P.R.) 1994 Ph.D Thesis, Department of Mechanical Engi¬
neering, Victoria University of Manchester. Methods of analysis for
flight flutter data.
[2] Fei (B.J.) 1984 Rapport de stage de fin d’ etudes, ISMCM, St Ouen,
Paris, France. Transformees de Hilbert numeriques.
[3] Haoui (A.) 1984. These de Docteur Inginieur, ISMCM, St Ouen,
Paris, France. Transformees de Hilbert et applications aux systemes
non lineaires.
[4] King (N.) & Worden (K.) 1994 Proceedings of 5^^ International Con¬
ference on Recent Advances in Structural Dynamics. An expansion
technique for calculating Hilbert transforms.
[5j King (N.E.) 1994 Ph.D Thesis, Department of Engineering, Victoria
University of Manchester. Detection of structural nonlinearity using
Hilbert transform procedures.
[6] Press (W.H.), Flannery (B.), Teukolsky (S.), & VetterHng (W.) 1986
Numerical Recipes, The Art of Scientific Computing. Cambridge Uni¬
versity Press.
[7] Simon (M.) 1983 Ph.D Thesis, Department of Engineering, Victoria
University of Manchester. Developments in the modal analysis of lin¬
ear and non-linear structures.
1677
[8] Simon (M.) & Tomlinson (G.R.) 1984 Journal of Sound and Vibration
90 pp. 275-282. Application of the Hilbert transform in modal analysis
of linear and non-linear structures.
[9] Stewart (I.) Tall (D.) 1992 Complex Analysis Cambridge University
Press.
[10] Worden (K.) & Tomlinson (G.R.) 1990 Proceedings of the 8*^ Interna¬
tional Modal Analysis Conference, Orlando, Florida pp. 121-127. The
high-frequency behaviour of frequency response functions and its effect
on their Hilbert transforms.
1678
Numerator Roots
-1214.211008298219 - 726.1763609509598i
-198.1314154141764 + 30.36502191203071i
-183.7196601591438 + 0.07476628997601653i
-173.0468329728469 - 0.58231241517227382
-167.4159433485794 + 0.24993126862275962
-148.7724514912166 - 0.5776974464158378i
-106.029245566151 + 6.329765806548005i
-105.755520052376 - 6.4400713086431752
-103.9415121027209 - 0.22926381395787662
-100.8868331667942 + 13.96371388356606i
-100.3491120108979 - 13.55839223906875i
-65.16772266742481 - 0.01242815581475099i
-1.161302705005027 x 10“^° + 1357.5723892932912
65.16772266742481 - 0.012428155814750992
100.3491120108979 - 13.55839223906875i
100.8868331667942 + 13.96371388356606i
103.9415121027209 - 0.2292638139578766i
105.755520052376 - 6.440071308643175i
106.029245566151 + 6.3297658065480052
148.7724514912166 - 0.5776974464158373i
167.4159433485794 + 0.2499312686227509i
173.0468329728468 - 0.5823124151722609i
183.7196601591439 + 0.07476628997601194i
198.1314154141764 + 30.365021912030712
1214.211008298219 - 726.1763609509598i
Table 1: Polynomial Numerator Roots for HCSl data
1679
_ Denominator Roots _
-198.1567099579465 + 30.40483782876573i
-183.7196564899432 + 0.07470973423345939i
-173.0468138842385 - 0.58222324003507392
-167.4160993615824 + 0.24976877099244062*
-148.7725565604598 - 0.5774140178292062*
-106.1570440779208 - 5.3529817854685192
-105.7091945278726 + 6.502652089082012
-103.941962628465 - 0.2315241290997767i
-102.9807367369542 - 9.760731644734012
-100.121488863284 + 14.02519387639362
-97.6024028604877 - 15.116504458164732
-65.16768960663572 - 0.012732667738438052
65.16768960663572 - 0.01273266773843805i
97.6024028604877 - 15.116504458164732
100.121488863284 + 14.02519387639362
102.9807367369542 - 9.760731644734012
103.941962628465 - 0.2315241290997704i
105.7091945278726 + 6.502652089082006i
106.1570440779208 - 5.3529817854685232
148.7725565604598 - 0.57741401782920622
167.4160993615824 -I- 0.24976877099244132
173.0468138842385 - 0.5822232400350746i
183.7196564899432 + 0.07470973423345935i
198.1567099579465 + 30.40483782876573i
Table 2: Polynomial Denominator Roots for HCSl data
1680
0.0000 Frequency / Hz 38.400
Figure 1: Bode plot of low exitation Duffing oscillator FRF.
J _ ! _ ! _ 1 - 1 - 1— - 1 - ! - ^
-500. 00U Real / mjN +300. 00u
Figure 2: Overlay of RP model Grp{u}) and original FRF for low
excitation Duffing oscillator.
1681
Figure 3: Original G{uj) and RP Hilbert transform Grp(u}): Duffing oscil¬
lator - low excitation.
Figure 4: Comparison of RT and FFT Hilbert transforms for low excitation
Duffing system.
1682
Figure 5; Overlay of RP model Grp{uj) and original FRF G{oj) for high
excitation Duffing oscillator.
Figure 6: Original G(w) and RP Hilbert transform Grp{u}): Duffing oscil¬
lator - low excitation.
1684
Fractional Fourier Transforms and their Interpretation
D.M. Lopes, J.K. Hammond and P.R. White
Institute of Sound and Vibration Research
University of Southampton,
Southampton SOI 7-1 BJ
United Kingdom
Abstract
Fractional Fourier Transforms arise naturally from optical signal
processing. However, their interpretation for one-dimensional signals
are still to be fully explored. This paper is concerned with fundamental
aspects, including :
• Definition and relation to optics
• Interpretation of signal decomposed in terms of 'linear chirps'
• A generalised concept of group delay
• The relationship to time-frequency distributions
• Analytic and computational examples showing the relevance
of the decomposition for non-stationary signals occurring
commonly in acoustics
Introduction
The Fourier Transform (FT) is perhaps the most widely used
tool in the signal processor's armament. It's simple definition, both in
the analogue and digital domains enables fast and efficient
computation. However, we should sometimes consider whether the
information given to us via the FT, could not be better represented in
some other form. The Fractional Fourier Transform (FRFT) is a
generalisation of the traditional FT, which by changing a given
parameter in the transform allows us to continuously move between
the time and frequency domains. First used for solving fractional
differential equations, the transform later found use in optical physics,
and this link is explored in the first section. It can be shown that the
FRFT can be viewed as a transform which has a set of basis functions
consisting of 'linear chirps'; this is covered in the second section. These
are simple harmonic waves, whose frequency changes linearly with
1685
time. In the third section we concern ourselves with the question :
'Given a signal, at what time does linear chirp make its maximum
contribution?' This is a generalisation of the work done by Zadeh [9]
for the traditional Fourier case. We then investigate the link between
the FRFT and Time-Frequency Distributions (TFD). The link between
the FRFT and the Wigner Distribution (WD) is explored. We complete
the paper by illustrating the above work using an analytical and
numerical examples of the manner in which the FRFT decomposes
signals.
The FRFT and its Relation to Optical Physics
The simplest way of visualising the form of the FRFT is to
consider it in terms of simple optical physics. Figure 1 shows such an
example of a simple projector type system. Ozaktas [5] was the first to
view the FRFT in this form, and started transition of the transform
from pure mathematics into more general usage. Even from this
simple system, it is useful to define a few key points. The input image
into the system can be considered to be the resultant image produce
immediately behind the lens, labelled a=0 on the figure. As we move
further away from the lens the image converges at the focal plane, this
is labelled on the figure as a=;r/2 [1].
At the focal plane the image produced is the 2-D Fourier
transform of the input image. If the input image is constant coherent
light across the whole of the input image plane, then at the focal plane
we would expect to simply see a point of light. This is because the
Fourier Transform of a constant is a delta function, as in this case we
are dealing m two dimensions it therefore follows that we should see a
'spot' in the centre of the screen.
As we move past the focal plane the image begins to expand
once more. At a certain distance we will recover a scaled version of the
original image, this point is labelled 0i=7r. At this point we have
applied two Fourier transforms to the input image, resulting in a scaled
version of the input. This is as expected from the theory of the Fourier
transform. We have therefore seen that the Fourier transform of the
input signal is at a distance from the lens of a^n/2 and that the
operation is repeated at a distance of a-K. At distances from the lens,
which are not multiples of k/2, the resultant image is a fractional
Fourier Transform of the input image.
These optical properties are stated in mathematical terms in [2-
3]. A given transform (F“) can be considered to be a FRFT, if it
1686
conforms to the a set of properties as outlined in Almeida's paper [4].
(Where I is the identity operator)
1. Zero Rotation : I
2. Consistency with the Fourier Transform : FT
3. Additivity of Transform :
4. Periodic : I
Although these properties define the FRFT completely, they do
not uniquely define it. However it has been shown that the majority of
the definitions given in the literature are consistent.
Linear Representation of the FRFT
Almeida [4] defined such a FRFT as follows :
oo
XJu) = J x{t)K^(t,u)dt ^ x(t) = J X„{u)K.^{u,t)du (1)
Were K.oit,u) is the kernel of the FRFT as is defined as :
ll-JcotCa)
\ 2k ^
K^{t,u) = l 6(t-u) a = 2nK neZ
5(t + u) a + K~2nK
1687
Where t is the time variable, and u is the generalised frequency
axis (i.e. (o as a-47u/2). Almeida approached the FRFT in much the
same manner as other authors on the topic. Starting from the basic
properties of the FRFT, he expressed the FRFT in a form similar to that
expressed by Ozaktas [5]. Almeida stated that the FRFT consists of
expressing x(t} on a basis formed by the set of functions defined by the
kernel (u,t). The basis is orthonormal since it can be shown that :
Bl{u ,t) dt = S{u-u!) (2)
The basis functions are 'chirps', i.e. complex exponentials with
linear frequency modulation. The FRFT can now be viewed as the
decomposition of the signal in terms of linearly changing harmonic
waves or 'chirps', sweeping at a rate dependant upon the parameter a.
This can be seen to be consistent with the definition of the FRFT as a
generalised FT, since in the limit as the parameter a-»7iy2, the linear
chirps become simple harmonic waves.
We also can consider the FRFT to be a rotation of the TF plane.
If a=0 then we are along the time axis and thus the input signal
remains imchanged. If a=;z/2, then we are on the frequency axis, and
thus the FRFT should be precisely the same as the FT. At rotations
which are not multiple of ;r/2, we are then in a domain which is neither
time nor frequency but a mixture of both.
The parameter a can be considered to be the chirp rate, that is
the rate at which the swept sine waves of the basis functions increase in
frequency. As a^7i/2 the basis functions revert to the simple harmonic
waves, as a reduces down to zero, the rate at which the chirps change
in frequency increases. It can be shown that as a-^0 that a chirp in
Eqn. 1 can be viewed as a Delta function [10, p.99]. The definition of
the transform as written in terms of the kernel function as defined by
Almeida and Ozaktas then becomes consistent with the viewing the
FRFT as a rotation of the FT plane.
Almeida's linear representation of the FRFT as a decomposition
of a signal in terms of linearly modulated harmoiuc waves allows an
intuitive hold upon the transform, and can be seen to be of vital
importance over the next few pages.
1688
The Generalised Concept of Group Delay
The idea of Group Delay (GD) was well explained by Zadeh, [9,
p. 441-444] the core of his idea is as follows. In it's traditional
definition GD is the amount by which a given (very small) band of
frequencies make their maximum contribution towards a signal.
Zadeh was able to show that the GD of a given signal could be
expressed as follows :
A ^ sin
cos{coj-^{co„„))
where A is the bandwidth of the group of frequencies under
consideration, and 0' = — — . This can be viewed as a sum of cosine
dO)
waves, modulated by a sinc(x) type function, which has its centre (and
maximum value) at - q' . This means that by finding 0' we can find the
time at which a band of frequencies make their maximum contribution
to the signal.
A classical example of group delay can be seen if we introduce a
pure delay into the input signal. A pure delay of a signal introduces a
phase shift in the FT as follows :
The complex exponential does not change the amplitude of the
FT, but introduces a linearly changing phase whose negative gradient,
-0', is proportional to the delay introduced into the signal. It thus
follows that the derivative of the phase, -0', is constant, and is the
amount by which the signal has been delayed. This serves to reaffirm
the definition of group delay, since we defined -0' as the time at
which a band of frequencies make their maximum contribution, which
in this case is the amount by which the signal has been delayed.
We now show that is it possible to extend Zadeh's original
definition using the FRFT, to answer the question : At what time does a
'chirp' of rate a, with centre along the u axis make it's maximum
contribution to the signal? Starting from our original definition of
taking the inverse transform back into the time domain and writing
the transform in terms of an modulus and phase it follows that :
1689
- cot(a)-a;csc(a)-0„(w)
2k
du
By dividing up the u axis into equal segments of length A. Then
(3)
x(t) =
l-ycot(a)
(n+I)A
2k
X J 4(“)-«
— — cot(a)-Kfcsc(a)-0„(«)
du (4)
n=-oo
If we make A sufficiently small so that for all segments Aa is
constant, we can make the following re-arrangement, defining the
centre of the bandwidth as +-^jA :
l-Jcot(a)
'“■’I— s—
(n+l)A
X j
n=-oo ^
j\ - co[(Qr)+///csc(a)-(>„(i;)
du (5)
Furthermore if we require that A be sufficiently small that over
each internal nA..(n+I)A we can use the Taylor expansion for (Pq(^)^
centred around giving
(l)„iu) = (l)^(u^J + (u-u„J
du
By defining <})■ = , and making the above substitution into Eqn. 5
we can write
l-Jcot(a)
X J ^
- cot( a )+ut csc(a )-0„ ( u„„ )-( u~u„„ )0’
du (6)
n=—oo
After a little manipulation of the exponential it is possible to show :
1 _ /cotfa) -y[ ^cot(a)+0„{u„„ )-«„„/ csc(a) "»»!’^ y— !^cot(a)+(fcsc(a)-6')(«-«,
LK n=-oo
(7)
The integral can be expressed in terms of Error functions, and it
is then possible to write Eqn. 7 as follows:
1690
^(0 =
/l-;’cot(ay 42
It: y J cot(a)
n=-oo
_J|r|^^i^l!^+csc-(a)j+f(-u„„c^*(a)-2u„„csc(a)a)l(a)-20\*sc(a>)-^>,,{n,,J+2(5'w„„c(^t(a)+(^'- Erf[l]]
(8)
Where Erf [1], [2] are defined as :
' J 42 {-2u„„ cot(a) - A cotfa) + 2t csc(a) - 20' ) ^
Erfm = Erf
Erf [2] = Erf
4^;cot(a)
7V2(-2m„„ cot(a) + Acot(Q:)-H2rcsc(a)-20')
44Jcot(a)
)
Although somewhat complicated the result can be considered to
be a generalisation of Zadeh^s result, as a-^7c/2, the above result
simplifies down to Zadeh^s result. Loosely the Erf terms can be
considered to be the general form of the sine function used in the
traditional Fourier case. The parameters u^n and b' shift the centre of
the function, A effects bandwidth, and a tightens the function about
the centre. As with the Zadeh definition, we now wish to find the
maximum value of the modulating function, to determine when the
chirp makes it's maximum contribution. By elementary mathematics
this can be found as :
t = cos(a) + 0' sin(a)) (9)
This can be checked against Zadeh's result by setting the sweep
rate to a=%/2. The result is interesting, because of the underlining
simplicity hiding the complex mathematics preceding it. Stated again,
this can be seen as the maximum time at which a linearly swept sine
wave makes its maximum contribution to a given signal.
The FRFT and it's Relation to TFD's
The concept of GD (and instantaneous frequency) allows us to
view the internal structure of analytical signals. However, for non¬
stationary or time-varying signal we need some other method to view
1691
the changing frequency components of a signal (i.e. [1]). The use of
Time-Frequency analysis allows us to construct a TF plane, allowing
the visualisation of the signal in terms of time and frequency.
Almeida's most significant contribution to the field of FRFT was to link
the FRFT and Time-Frequency analysis. Almeida viewed the FRFT in
terms of a rotation of the Time-Frequency Distribution (TFD) and the
application of a Fourier Transform. More important still was the link
that Almeida established between the Wigner Distribution (WD) which
is one such method of constructing the TFD and the FRFT.
After a little mathematics (see [4]) Almeida's deduced that the
WD could be written as ;
Where u = t cos a + co a , and v = - ? a + © cos a
The left hand side of this equation is the WD of Xo[ u) computed
with the arguments (r,©), and the right is the WD of x(t) computed with
arguments (m,v). The equation shows that the WD of Xa is the same as
the WD of X if we take into account the rotation due to the change of
variables.
" This is equivalent to saying that the WD of Xa is the WD of x,
rotated by an angle of -a, or that it is simply the WD of x expressed in
the new set of co-ordinates Almeida, [4, p. 3088]
Lohmarm and Soffer's paper [6] shows essentially the same
property, but use the Radon transform to produce the required
rotation and projection. Since Almeida published his paper, other
authors have established this link not just for the WD, Ozaktas et al. [7]
showed that the same properties were true for all members of the
Cohen class which are rotationally symmetric about the origin.
Another way of viewing the FRFT is as a rotation and projection
of the TF plane. Figure 2 shows this idea. We can assume that a linear
chirp is simply a straight line in the TF plane. If we rotate the time and
frequency axis by the sweep rate of the chirp, and project along the u
axis, we have constructed the FRFT. This is an intuitive representation
of the mathematics given by Almeida and Ozatkas. This rotation and
projection is called the Radon transform, and in [6] the presentation of
the FRFT as a Radon transformed WD was shown.
1692
Figure 2 - FRFT is terms of rotation and projection.
Frequency
Examples
1. Rectangular Function
The first of our examples, shows the analytical decomposition
for a rectangular function. The FRFT can be written in terms of Fresnel
integrals, as expressed below :
’ 2cot(a)
y
2
(‘■4)
K
( 2b|5-r^
-K
^2ap~y^
[ VW ;
l^/W JJ
(11)
Where K is defined as the Fresnel integral :
X jm~
K{x) = \e - dz (12)
0
This function is shown in Figure 3. The function can be seen to
evolve as the sweep rate is changed. As a— >0 we can seen that the
FRFT tends towards the rectangular function as expected. As the
sweep rate increases the function develops, with oscillations in the
transforms becoming slower. As a“»7c/2 the function develops into the
a sine type function, the Fourier transform of a rectangle.
1693
2. Doppler Type Signals
The acoustics of a moving sound source are extremely
complicated, both the shift in frequency (Doppler shift) due to the
movement of the source, and the propagation delay of the speed of
soxmd must be taken into accoxmt. In the following example, we
simplify the situation by considering the Doppler shift without
propagation delay. In order to simulate the change in level of the
sound experienced due to the distance of the vehicle from the
microphone, we will use a Gaussian function to reduce the signal to
zero at the beginning and end of the signal.
If the source and receiver approach each other at speeds of v
and u respectively, and the source emits a tome frequency of f„ when
stationary, then it is possible to show ([8], p. 415) that the receiver will
hear a frequency/’ where : (c is the speed of sound)
/ = /o— (13)
c-v
This change in frequency is known as the Doppler Shift. A schematic of
the type of system we are trying to model is shown in Figure 4.
If the microphone is stationary, and the car moves passed it at
constant speed, then a modified form of the equations [8] describes the
frequency received by the microphone (the signal is shifted in time in
order to keep the equations causal) :
f = fo
1 + -
VCOSI
m'
(14)
1694
were 0(f) =
-tan
;r-tan
v(f-lO)
v(r-lO)
f < 10
f>10
This is a non-linear function; however as the car approaches the
microphone, it becomes almost linear. Figure 5 shows the changing
frequency component with time for two different (simulated) vehicle
speeds. (The curve for the faster speed is marked with signs.) We
will assume for the sake of simplicity, that the vehicle emits only one
tone, and that it travels at constant velocity.
1695
The greater the velocity of the vehicle, the steeper the curve.
The angle, Q, varies from Q..k. The rate of decrease of frequency can be
found, and is shown to be
dt
xL sin
tan'
v(r-lO)
c{t-\0)-
1 + —
v'(r-lO)'
(15)
Since the rate of change of frequency is almost linear at the point
when the vehicle passes the microphone, application of the FRFT (at
the correct sweep rate setting) will produce a strong spike like
distribution. From the knowledge of the sweep rate and thus the rate
of change of frequency, it is possible to infer the velocity of the vehicle.
Using this simple model of a Doppler shifted signal, and making some
basic assumptions (the observation point i.e. microphone is stationary,
constant velocity, and single pure tone emission of the source) we
simulate the situation. The synthesised signal is shown in Figure 6,
and its Spectrogram is depicted in Figure 7.
From the synthetic signal the FRFT is used to find the sweep
rate which best matches that of the input. Thus is we pick the right
sweep rate, the chirp produced by the vehicle will show up as a spike,
or an least a peak in the distribution. In our example case were we
know the input parameters,
1696
it is possible to predict at what sweep rate, and were on the u domain
we expect to see a rise in the distribution. In this example case it has
been possible to predict the sweep rate and position of peak on the
axis, however in a real situation this would not be the case. Multiple
transforms would have to be taken on the data, possibly in some
iterative way to find the peak in the distribution. However due to the
nature of the delay equations, and the low sweep rate, the location of
the peak will not change very much from one velocity to another.
The result of the application of the FRFT can be seen in Figure 8,
the peak which was expected does not stand out greatly from the rest
of the data because of the way in which the signal changes. Close to
the region we are considering to be a linear chirp, other almost linear
chirps are also present. These other chirps spread out the distribution
into the form seen in Figure 8.
From the knowledge of the rate of change of frequency, it is a
simple manner to calculate the passing velocity of the vehicle.
1697
Frequency
Figure 7 - TF Plane of input signal.
Sin(Phi)
Time
1698
Conclusions
In this paper we have covered some of the theory and applications of
the fractional Fourier transform. We have shown that the fractional
Fourier transform can be considered to be a generalised version of the
Fourier transform, having a basis of swept sine waves rather than
harmonic waves. We have considered the notion of ^group delay' as
being the time and which a band of frequencies, and generalised this
idea to the form where by we can answer the question 'At what time
does a chirp, of a given sweep rate make it's maximum contribution to
a signal ?' The link between the FRFT and TF has been explored. We
have investigated some examples of signal using this techmque. Finally
we have outlined one application of the FRFT as a way of determining
the degree of shift in a Doppler type signal.
References
1) D. Lopes 1996 Disseration submitted as part of an M.Sc. in Soimd
and Vibration, at ISVR, University of Southampton.
2) A.C. McBride and F.H.Kerr 1985 IMA Journal of Applied Mathematics
39, 159-175. On Namias's Fractional Fourier Transforms
3) S.Abe and J.T.Sheridan 1994 /. Phys. A: Math. Gen. 27, 4179-4187.
Generalization of the fractional Fourier transform to an arbitrary linear
lossless transform: an operator approach
4) L. B. Almeida 1994 IEEE Transactions on Signal Processing 42(11),
3084-3091. The Fractional Fourier Transform
5) FI.M.Ozaktas and B.Barshan 1994 /. Opt. Soc. Am. A. 11(2), 547-561.
Colvolution, filtering, and multiplexing in fractional Fourier domain
and their relation to chirp and wavelet transforms.
6) A.W.Lohmannm and B.H.Soffer 1995 J. Opt. Soc. Am. A 11(6), 1798-
1801. Relationships between the Radon-Wigner and fractional Fourier
Transform
7) H.M.Ozaktas, N.Erkaya and M.A.Kutah 1996 IEEE Signal Processing
Letters 3(2), 40-41.Effect of Fractional Fourier Transformation on Time-
Frequency Distributions Belonging to the Cohen Class
8) L.E.Kinsler and A.R.Frey 1982 Fundamentals of Acoustics. New York :
John Wiley and Sons Inc.
9) L.A. Zadeh and C.Desoer 1963 Linear System Theory. New York :
McGraw-Hill.
10) A. Papoulis 1984 Signal Analysis. New York : McGraw-Hill.
1699
1700
SYSTEM IDENTIFICATION III
WAVE LOCALIZATION EFFECTS IN DYNAMIC SYSTEMS
J. Dickey
Center for Nondestructive Evaluation,
The Johns Hopkins University, Baltimore, MD 21218, USA
G. Maidanik
David Taylor Research Center. Bethesda, MD 20084, USA
J. M. D'Archangelo
United States Naval Academy. Annapolis, MD 21402, USA
Abstract
Structures with discrete periodic variations in impedance may exhibit pass
and stop bands and the related wave localization and delocalization phenomena
in their frequency response. Localization, similar to Anderson localization in
atomic systems, occurs in the pass-band frequency range when the periodicity
is perturbed and waves are thereby inhibited from propagating. Conversely,
delocalization occurs in the same systems in the stop-band regions where
perturbing the strict periodicity allows for relatively more propagation.
Localization and delocalization are demonstrated in several systems:
specifically, a beaded string, membranes and plates with periodic stiffeners
attached, and a "jungle gym", i.e. a connected beam structure. It is
demonstrated that these effects depend on the interactions between
discontinuities.
Introduction
A structure with strictly periodic impedance discontinuities exhibits pass and
stop bands in its frequency response. The pass bands are frequencies at which
waves traverse the structure relatively unimpeded and thus give rise to modes
which span the entire structure: i.e. global. Stop bands are frequencies at
which waves are impeded from traversing the structure and give rise to modes
which are confined near the source: i.e. local. When the strict periodicity of
the structure is disturbed by. e.g., randomly varying the distances (or
equivalently, the wave propagation speed) between discontinuities, the pass-
1701
and stop-band character in the frequency response is , in some degree, spoiled.
The "spoilage" of the strict pass-band behavior results in the long range wave
propagation becoming shortened with the attendant localization of the modes.
This effect is called localization. Similarly, the "spoilage" of the strict stop-
band behavior results in a lengthening of the wave propagation and a
delocalization of the modes. This effect is called delocalization. Both
localizing and delocalizing cause the long and short range wave propagation,
respectively, to tend towards a middle ground, called a "yellow" band (amber
to some), where waves are neither short nor long and where modes are neither
global nor local. It is proposed that delocalization is as important a phenomena
as localization with relevance to practical applications.
These phenomena occur in a wide variety of dynamic systems ranging from
atomic lattices to rib-stiffened ship hulls.^’-*^^ In fact, the phenomenon of
localization in atomic systems was popularized by P. W. Anderson in 1958
who characterized it as "absence of diffusion in certain random lattices
Anderson was working with the wave equation describing the electron density
in crystalline lattices and noted that randomizing the spacings of an originally
regularly spaced crystal tended to inhibit the diffusion of electrons from a point
of injection. The analysis here is limited to macroscopic systems obeying
linear wave equations since the effects are fundamentally phase interference
and. as such, depend on superpositon of wave functions. This is not to say
that localization does not exist in non-linear systems, it is Just that the model
used to describe it here is limited to linear systems. Further, the analysis is
limited to one-dimensional systems in the sense that the wave equation for the
structure must be separable into functions of a single spatial variable which
contains a periodic discontinuity.
Both localization and delocalization depend on a spoiling of the strict
periodicity of a system. There are many ways to effect this "spoilage": one
which will be used extensively here and which will be used to establish a
benchmark for quantification of the phenomena, is to randomize the lengths of
the connected ID systems. The prototypical complex of connected ID systems
is shown in Fig. 1 and it is the length (b) of the constituent systems,
enumerated 1-N, which are to be randomized, i.e.
1702
= b(l + aRn) , 0<a< 1 ,
where Rn is a random number between -1 and 1, and a controls the degree of
randomization: e.g., a = 0.1 is referred to as 10% randomization and the
lengths are uniformly distributed, 0.9 < (ln)< 1-R The benchmark from
which to quantify localization and delocalization which is proposed here is the
limit which is approached when a system is completely randomized.
"Complete randomization" as used here means that a = 1 and if an ensemble
average is taken over a large number of realizations of Fig. 1, each with a
different set of Rn’s then the oscillatory nature of the phase change m the
amplitude of a wave traversing a system will average to zero. With the phase
effects averaged, the diminution of a wave traveling through the set of systems
will depend on the average propagation loss factor,
"■il, ”■ '
and the cumulative transmission coefficient,
T^(co) = n T„(cb) . (ic)
Without the ensemble averaging over realizations of Fig. 1, each with a
different set of Rn, the phase effects must be accounted for. This may be done
through the use of an impulse response matrix
g(x I x', Q)) = (gjjCXj I x'i , (»)) , (2a)
in the sense that it relates a response in system] at the coordinate location xj to
an impulse in system i at the coordinate location x'j . More generally, in linear
systems in which wave solutions superpose, Eq. (2a) is the Green function
kernel which relates the response at any point in any system,
v(x , co) = { Vj(Xj, co) } to a distributed drive at a particular frequency through
the usual relation;
1703
The impulse response matrix, Eq. (2a) is formulated for a connect set of 1-0
systems such as shown in Fig. 1 by following waves as they emanate from
the drive position x'j and propagate with multiple reflections and transmissions
at the junctions between systems/^)
Periodic Complex - Pass and Stop Bands
Equation (2) is used to calculate a response and to demonstrate the
phenomena of localization and delocalization in a set of dynamic systems such
as shown in Fig. 1. Certain parameters must be specified in order to do this:
specifically, the number of systems, their lengths, their impedances, and their
manner of attachment to each other. Accordingly, seven systems are assumed,
all stationary in time and all supporting a single, non-dispersive wave type
which propagates according to:
V, (Xj, CO) = V (x'j, CO) , (3)
The time dependent factor, e‘“^ is implicit in Eq. (3) and
kj =koj(l-irij)
koj = co/Cj
Cj = the phase speed of waves propagating in the j^h
system, generally may depend on frequency
(i.e. dispersive) but will be taken to be constant
here.
■q. =r lO”^, the propagation loss factor for the j^h
system
= the length of the jtk system, bis the unperturbed
length
mj = the mass of the bead.
The response, as calculated by Eq. 2 and Reference 5 for an infinite flexible
string with 8 equally spaced point masses (beads) attached is shown in
Fig. 2a. For the periodic complex all lengths are the same ^ j = b; also cj = c,
rij = Ti and mj = m. The complex is driven in the first bay as indicated and
the magnitude of the ratio of the response relative to the response at the drive
point x' is plotted as a function of position in the complex. The drive
frequency is (co/co^) = 10 where cOj is the fundamental resonance of the
unperturbed bay, i.e. cOj = 7tb/c.. The beads are concentrated masses equal to
0.3 times the mass of the string in the unperturbed bay. The standing wave
patterns in the bays are evident as is the exponential decrement of the response
as the excitation traverses the complex.
Figure 2a illustrates the exponential attenuation characteristic of wave
propagation through a cascade of connected one-dimensional systems. The
attenuation shown here is due to the combined effect of the propagation loss
factors of the individual bays (this is negligible here) and the phase interference
due to the reverberation within the bays. The degree of attenuation, i.e. the
slope of the log-response vs. distance curve is a function of frequency, and it
is this variation with frequency which gives rise to pass/stop band phenomena.
Accordingly, a series of responses of the type shown in Fig. (2a) are
calculated for a set of increasing frequencies and shown in Fig. (2b). The
frequency range encompassed by the calculation is 0 < (0)/c0|) ^ 20 and the
pass and stop bands appear as frequencies where the complex is transparent or
opaque respectively to the waves traversing the complex.
An alternate presentation of the response which shows the pass/stop band
phenomena more explicitly, is the response at the end of the complex,
essentially the transmissibility of the complex, as a function of frequency.
The plot in this format which corresponds to the parameters of Fig. (2b) is
shown in Fig. (2c). The peaks in the response shown here correspond to the
pass bands of the complex and are identical to the pass bands as identified in
Fig. (2b) as frequencies where the response shows minimal attenuation in x.
The minima in Fig. (2c) are the stop band frequencies which are more difficult
to identify in Fig. (2b) because, for clarity, the data shown there is
"shadowed" by curves in the foreground. The benchmark attenuation for the
1705
system discussed earlier which delineates between pass and stop band
behavior is shown by the dashed line in Fig. (2c)
A-Periodic Complex - Localization and Delocalization
The phenomena of localization and delocalization are demonstrated by
calculating the response of the complex where the lengths of the individual
systems are randomized. Accordingly, a system response with all parameters
the same as in Fig. (2) except that a = 0.2 in Eq. (la) is shown in Fig. (3).
Note the dismption of regularity in the decrement as the excitation traverses the
systems. The transmissibility shown in Fig. (2c) for the regular complex is
repeated for the randomized one and shown in Fig. (3c). It should be kept in
mind that the calculations shown in Fig. (3) are for a specific realization of
randomized lengths and would not be exactly the same for a different
realization of the random number set used to generate the lengths; different
realizations, however, show similar behavior. The dashed line in Fig. (3c),
again represents the benchmark attenuation.
The demonstration of localization/delocalization lies in the fact that the
peaks and valleys, respectively, of the transmissibility shown in Fig. (3c)
have moved toward the benchmark value. This is more true at higher
frequencies where the length perturbations represent a larger fraction of
wavelength.
Special Cases
Two more complex structures are considered. The first is an infinite
membrane with an array of 8 stiffeners or ribs attached as shown in Fig. 4a.
In the unperturbed case, the rib spacings are all equal and of length b and the
ribs are modeled as line impedances. The structure is driven and the response
is assessed along lines parallel to the ribs at positions x' and x respectively, as
shown. It should be noted that the pass/stop band and localization phenomena
do not depend strongly on the magnitude or nature of the rib impedances; i.e.
they can be mass-like, resistive, spring-like, resonant, etc. The pass- and
stop-band characteristics for this structure where the rib impedances are mass¬
like with a mass equal to 0.3 times the mass in a bay of length b are shown in
Fig. 4b. The solid curve is the transmissibility vs. frequency of the periodic
1706
(unperturbed) case, and the dashed curve is the aperiodic (perturbed) case with
a = 0.2. Again, the normalizing frequency co, is the fundamental resonance
of the system of length b. It should also be noted that if a thin plate were used
in place of a membrane, the data shown in Fig. 4b would be similar but show
a progressive widening of the frequency separation of adjacent bands with
increasing frequency due to the dispersive nature of the plate free-wave
propagation. The membrane was given a propagation loss factor of
q = 5 X 10"^ in the direction of propagation and the slight progressive
decrement of the pass-band peaks with frequency reflects this; i.e. the structure
would exhibit unity transmission at the center of the pass bands if there were
no loss.
A second example of locaiization/delocalization effects in more complex
systems is the planar array of connected one-dimensional systems shown in
Fig. 5a. The array dimensions are i3b x 7b where b is the length of the
unpermrbed system, and, for this example, the bead impedances were taken to
be zero. The drive and assessment positions were at x , = 0.3b and
= 0.5b respectively, as shown. The transmissibility of the regular
(unperturbed) complex is shown as a function of (co/co j) by the solid curve in
Fig. 5b and one realization of a perturbed case with a = 0.2 by the dashed
curve.
Conclusions
The phenomena of localization and delocalization is demonstrated in several
mechanical systems. The canonical system is an infinite string with a finite
array of beads attached in which wave localization and delocalization occur in
the pass and stop frequency bands respectively when the strict penodicity of
the bead spacing is disturbed. The phenomena are also demonstrated in a
membrane with a finite array of attached ribs. The final example is a two
dimensional, planar, array of beads connected with wave-bearing elements.
Strong localization and delocalization occur in this system also.
1707
References
[1] Hodges, C. H., 1982. Journal of Sound and Vibration S2, 411-424.
Confinement of vibration by structural irregularity.
[2] G. Maidanik and J. Dickey 1991 Acoustica 73, 119-128. Localization
and delocalization on periodic one-dimensional dynamic systems.
[3] Langley, R. S., 1995, Journal of Sound and Vibration, 188 (5), 717-
743, Wave transmission through one-dimensionai near periodic
structures: optimum and random disorder.
[4] D.M. Photiadis, 1996, Applied Meek. Reviews 49 (2), 100-125, Fluid
loaded structures with one-dimensional disorder.
[5] Anderson, P.W., 1958, Physical Review 109, 1492-1505. Absence of
diffusion in certain random lattices.
[6] L. J. Maga and G. Maidanik, 1983, Journal of Sound and Vibration, 88,
473-488. Response of multiple coupled dynamic systems.
0
System # 1, 2, etc.
• • • •
Beads
Fig. 1. The prototypical complex of coonected one dimensional systems, an infinite
string with seven eaually spa^ beads mtached.
1708
c)
Fig. 2. Periodic bead spaceing; a) the positions of the beads and the response as a
function of distance along the string at a single frequency, b) spatial responses as
in a) but at a series of frequencies, and c) the response at bead number 7 as a
function of frequency, the dashed line is the product sum of the transmission at
beads 1 to 7 inclusive and deleniates between pass and stop bands.
1709
c)
Fig. 3. A-periodic bead spaceing; in the same format as Rg. 2.
1710
b)
Rg. 4 a) An infinite membrane with 14 ribs attached, b) The tMsmissibility
as a function of frequency for the regularly spaced ribs (solid line) and for
one realization of a set with 20% random variation in rib spaceing (dashed
line).
Rg. 5 a) A 13 X 7 planar array of connected 1-D systems driven at x' in system
#1 ; b) the transmissibility as measured in system #202 as a function of frequency
for the regular system (solid line) and for one realization of systems with 20%
random variation in lengths (dashed line).
1711
Estimated Mass and Stiffness Matrices of Shear Building from
Modal Test Data
Ping Yuan, ZhiFeng Wu, and XingRui Ma
Department of Civil Engineering,
Harbin Institute of Technology, Harbin, 150001, P.R.China
ABSTRACT:
A method to estimate mass and stiffiiess matrices of shear building from
modal test data is presented in this paper. The method depends on that
measurable points are less than the total structural degrees of freedom, and on
first two orders of mode of structure are measured. So it is applicable to most
general test. By giving a method to estimate modal data of immeasurable points,
global mass and stiffness matrices of structure are obtained by using first two
orders of modal data. By use of iteration the optimum global mass and stiffhess
matrices are gained. Finally an example is studied in this paper. Its result shows
that this method is reliable.
l.Introduction:
Measurement of the dynamic characteristics, natural frequencies and mode
shapes of structure from modal dates is developed in recent years. Naturally,
there is only a finite number of points on the structure for which data can be
collected. These points are generally a small subset of the total degrees of
freedom(DOF) in a finite element model of the structure. In fact, the number of
measurement points may also be less than the total number of vibrational
modes identified in the test, specially when utilizing modem instrumentation
with high sampling rates and powerful, inexpensive scientific workstations for
data for n mode, where n>f there is not a unique model of classical mass and
stiffhess form with physical DOF that possesses order-n dynamics given only 1
spatial measurement points.
Much research in recent years has focused on methods for correlation or
reconciliation of finite element models that inherently possess very large-order
dynamics to the limited sensor and frequency data obtained from modal
testing.
The primary goal of the present paper is to investigate direct solutions to the
inverse vibration problem when the number of sensors / is less than the number
of identified modes n. We will show that mass and stiffhess matrices of
1713
dimension n, referring to the total DOF, can be found from the I identified
modes and the modal dates of the I points.
In this paper, the unmeasured modal dates are evaluated by the measured
modal dates. Finally, the mass and stiffhess matrices are obtain by first two
order modes.
Z.Evaluation of unmeasured modal dates
Recent work in the area of structural identification has included the
determination of mass and stiffiiess matrices directly from continuous time
system realizations. ^ This approach is that it requires the dimension of the
physical mode to be equivalent to the number of second-order states, implying
that the number of independent sensors measured are equal to the number of
identified modes. A more practical approach is to enrich the computer mass
and stiffness matrices with the complete set of measured modes, independent of
the number of sensors. This allows the resulting to express contributions of all
of the modes observable from the available sensors. We begin by developing
the concept of reduced the measured modal dates.
If structure is regard as shear building, we can assume that 1) the total mass
of the structure is concentrated at the levels of the floors; 2) the floors are
infinitely rigid as compared to the columns; and (3) the deformation of the
structure is independent of the axial forces present in the columns. These
assumptions transform the problem from a structure with an infinite number of
degree of freedom to a structure that has only as many degrees as floor levels.
Clearly the stiffiiess matrix of shear building is a triangular matrix, and the mass
matrix is a diagalization matrix. The mass and stiffhess matrices of shear
building can be written as:
^,+^z -K . ®
-k, fcj+fc, -fc, :
K= ••• (1)
-K-, K-,+K -K
0 . -K K _
~m^ ••• 0
M= : **. :
_0 -
High building always has standard floor levels. It is to said that these levels
have the same mass and stiffness. Assumed that the standard levels are i.
1714
i+1 . .j, j<n, where n is the total levels of building. In a word there are j-
i+l=d standard levels, and the frist level is not standard level. If i-1 and j levels
are placed sensors, these two points become measured points. We will use the
modal dates of these two points to evaluate the modal dates of unmeasured
points.
Suppose that we are given an arbitrary undamped MDOF system. The
differential equation of the system is given by :
MX+KX=F
where M is the structural mass matrix, K is the structural stiffness matrix. X is
the vector of generalized modal deflections, X is the vector of generalized
modal accelerations. F is the vector of external forces. Considered the
corresponding characteristic equation:
where is the /th eigenvalue, and (Ji^is the /th corresponding mode shape or
eigenvector. Eq(l) can be also written as:
(\kaa 0 1 rm„„ 0 0
K K. K. 0 rn,, 0 k =0 (2)
[[o K, /c„J [o 0
where a-l,2, . i-1, b=i,i+T . j-C c=j,j+l, . n, the mass matrices
m. . m are diaaanalization matrices, and the stiffness matrices can be written
00 ^ CC w
as:
"0 - -k-
0 0 j
■ 0 ••• 0~
and/c^, = : : =kl Q)
...
Eq.(2) is partition of Eq.(l). The second line of Eq.(2) can be expressed as:
^ba^a '^^bb^b ^oc ~^l^bb^b
This reduction concept considers the influence of mass. Substituting Eq.(3)
into Eq(4), we have:
0
Kb<^b - ^i^bb^b = ' f
0
1715
Expanding Eq.(5), we have:
(fc, +fci,,)<f),-
+(fc,>, +fc,V2 )<!),>, = 0
. (6)
)<!>;-, = <<J^J
Because from level / to level j are standard level, so their stiffness and mass
have following relationships.
m = m^- =•••= rrij
:ituting Eq.(7) into Eq.(6)
rr
aK} =
2-Xia -1
1 2-X,a -1
0
0 ••• -1 2-X^aj)
where a = j, {<t>6} = {(!>,, <!);-, using least
square approach the unmeasured modal shape } can be obtained:
<
> =2 <
0
>
'-K
-nO,-
-A,
l = D (8)
{(}),} = (A^rM'^'D (9)
By mean of Eq(9) those unmeasured DOF can be expressed by measured
DOF. It is to said those modal dates from i to j-1 can be evaluated by (i-l)th
and jth level modal dates. In this method the influence between the mass of
DOF and level displacement is considered. Using this method measured points
can be reduced.
3.The evaluation of mass and stiffness matrices
Expanding Eq(l) we have:
(kj+k^^i, - Mis =0
+(fc2 =0
. (10)
-K-,K-2 +<-K-, -fc.'t'to =0
In order to evaluate mass and stif&ess of structure, equation (10) can be
written as:
1716
(11)
-K^rl 4>r;-^r2
m,
^12 ^'^12 ‘1^/2"' ^13
^2
4>r2*‘J’r; V^r2 ^r2 “ ‘l>r3
m,
■ 1
'l>ln-7“'l^ln-2 ‘1^ In-; ~ *1*11,
:
:
K
<j^r75
where r is the rth mo(ie. This equation can be simply wntten as:
B{b} = 0 (12)
where {b} = . . In order to solve this equation the number
of equations should equal the number of unknown. Because the number of
equation is 2Xn, so we used two order modes. Clearly matrix B is a singular
matrix. In order to assuming rn^ = \, the Eq.(12) can be expressed as:
f 0 1
(13)
where B’ is the matrix that eliminate the last two lines and last two columns of
matrix B. {b’} is the vector that eliminate the last member of vector {b}.
Using least square approach (b’} can be obtained.
f 0 1
Mir,
(14)
The mass and stiffness matrices obtained by Eq.(14) are not the real value.
There is a constant rate between evaluate value and real value.
The following procedure has developed for evaluated the global mass and
stiffness matrices of shear building.
Step 1: Choose the initial parameter
a is the ratio between the mass and the stifhiess of standard levels. We
1
defined the initial parameter = — .
Step 2: Evaluate the mode shapes of unmeasured DOF
In this step Eq.(9) is used to estimated the two order mode shapes and
. Where and |(j)} are the /th and rth corresponding mode shapes.
1717
Table 2.
ki(N/m)
k2
ks
k4
mi(kg)
m2
nu
Real value
18000X
6
18000
X3
18000
X5
18000
X2
18000
X2
18000
X2
Evaluation
2.9948
0.99896
0.99896
1.0
5. Conclusion
A new method to evaluated mass and stiffness matrices of shear building
from modal test dates is presented in this paper. This approach based on the
number of measurement points less than the number of total degrees of
freedom of structure. In this method only first two order modes are used and
the influence of mass is considered. By mean of this method it is possible of
using modal test dates to evaluate the global mass and stiffhess matrices and
the damage of structure.
6. References
1. Kabe,A.M. Stiffenss Matrix Adjustment Using Mode Data. AIAA. 1985.
Vol23. No.9. 1431-1436
2. Farhat,C., and Hemez, F., Updating Finite Element Dynamic Models Using
an Element-By-Element Sensitivity Methodology. AIAA, 1 993, Vol 31,
No.9. 1702-1711
3. Yang,C.D., and Yeh,F.B., Identification, Reduction, and Retinement of
model Parameters by the Eigensystem Realization Algorithm. J. of Guidance,
Control and Dynamics, Vol. 13, No. 6, 1990, 1051-1059
4. Baruch, M., Optimization Procedure to Correct Stiffness and Flexibility
Matrices Using Vibration Tests, AIAA, Vol. 16, 11, 1978, 1209-1210
5. Topole,K.G., and Stubbs,N., Non-destructive Damage Evaluation of a
Structure from Limiten Modal Parameters, Earthquake eng. struct, dyn., Vol
24, 11, 1995, 1427-1436
1718
THE PROBLEM OF EXPANDING THE VIBRATION FIELD FROM
THE MEASUREMENT SURFACE TO THE BODY OF AN ELASTIC
STRUCTURE
Yu.LBobrovnitskii
Blagonravov Institute of Mechanical Engineering Research of the Russian
Academy of Sciences, Moscow 101830, Russia
Introduction
Knowledge of the distribution and magnitude of the dynamic stresses of
an engineering structure due to extensive vibration is important for the
estimation of the structure reliability and its mean service time. Another
practical problem where it is needed is control of the structural vibration:
knowledge of the stresses and displacements allows one to compute the vector
field of the vibration power flow which makes visible the sources and
transmission paths of vibration and thus indicates adequate means to control it
[1,2].
Commonly used sensors, e.g., accelerometers or strain gauges, can
measure the vibration parameters (acceleration or strains) in discrete points.
Some instruments developed in the last decade such as laser interferometers [3]
and the vibrometers based on the near field acoustic holography [4], can
measure continuous distribution of the vibration amplitudes at a certain surface.
Nevertheless in practice, at best only a part of the structure surface is accessible
for measuring vibration directly. For most of the interior points of the structure
body as well as for some portions of its surface, mounting sensors is impossible
or impractical. So, the only way of estimating the stress-strained state of the
whole structure is to expand the vibration field from the measurement surface
to the structure body.
In this paper, an approach to such expansion is proposed which is
applicable to complex elastic structures of which a part of the surface is
accessible for direct vibration measurements while the rest of the surface and
the body are not. This approach consists of measuring the distribution of the
amplitudes for three components of the vibration displacement (or acceleration)
on a portion of the accessible surface. For the volume contiguous with this
portion, a special boundary-value problem which is called here the problem of
field reconstruction (FR-problem) is stated and solved (see Fig.l). As a result,
the displacement (and stress) field of this volume is determined (reconstructed)
through the measurement data. Then, using the vibration amplitudes measured
1719
on another portion of the
accessible surface, one can
similarly determine the
displacements and stresses in
another volume, etc., until
the whole structure is
investigated.
The idea of
reconstructing the general
pattern from incomplete or
indirect data is not new and
is often used in mechanics
and structural dynamics. It is
used, for example, in the
mode shape expansion
methods of the model
updating techniques, where
the responses measured on a
part of the DOF’s of the FE-
model are expanded to the slave (unmeasured) DOF’s using the equations of
motion [5], Another example is reconstruction of the time history and
localisation of external forces from the measured structural responses [6].
All the cases, where the idea of reconstruction is realised, are distinguished by
the statement of the corresponding mathematical problem and by physical
peculiarities. In this paper, the key problem is stated as reconstruction of the
vibration field in a finite elastic solid from the amplitudes of the displacements
of a part of its boundary free of tension. In structural dynamics, such statement
was first proposed in Ref [7] relating to measurement of the vibration power
flow in solids. Similar statements are met in a number of papers on the static
theory of elasticity (e.g., [8]) for reconstruction of the stresses inside a machine
part through the measurement of the strains on its surface.
In what follows, the rigorous mathematical formulation of the FR-
problem is given and its general properties are studied. Results of the computer
simulation and of the laboratory experiment are presented which prove the
practicality of the approach suggested.
u(s)=uo f(s)=0 (b)
Fig. l.An elastic structure (a) of which only the
upper surface S is accessible for vibration
measurement; (b) boundary value problem for
the chosen volume V: displacements u and
force f are specified on S.
Formulation of the problem
For a selected elastic volume V of the structure (Fig. lb), the problem of
the harmonic field reconstruction can be formulated as follows (time
dependence exp(-icot) is omitted);
find a solution to the homogeneous Lame equations
1720
(1)
^JhM(x}+{X+|ijgxz.&.Av^u(x)+p(^u(x)=■0, x&V,
which satisfies the conditions
u(s) = uo(s), f(s) =0, s eS, (2)
on the upper part S of the surface. Here x= (xj, X2, X3) are the co-ordinates of a
point of the volume V, u = (ui, U2, U3) is the displacement vector, f=(f},f2,
fs) is the vector density of the forces acting on the surface, X and /i are the
elastic Lame coefficients, and Uo(s) are the known (measured) displacement
amplitudes.
The formulation (1),(2) is not traditional for equations of the elliptic
type: the boundary conditions on the part S of the surface are overdetermined,
i.e., both the displacements and the forces are specified, whereas no quantities
are specified on the rest part 0 of the surface.
The boundary- value problem (l),(2) may be formulated as an integral
equation. Let G(x/q) be the 3x3 matrix of the Green's functions describing the
response (displacement vector) at a point x e F to a unit force at a point q e
0. Assuming x = s e S, the problem (1),(2) may be rewritten as
uo(s) =ffG(s/q)f(q)dQ. (3)
This is a set of three Fredholm equations of the first kind, where the unknowns
are the reaction forces f(q) acting on the inaccessible surface 0; the matrix
G is assumed to be known. If the forces f(q) are found from Eq.(3), then the
displacement field of the volume V can be computed as
u(x) =f/G(x/q)f(q)dQ. (4)
Q
The formulations (1),(2) and (3),(4) are convenient for investigating the
general properties of the problem. In practice, preferable is the formulation
based on the expansion of the field in the normal modes of V
u(x) =Za„q>Jx), (5)
where the forms q)„(x) are assumed to be known and the amplitudes a„ must
be determined from the first boundary condition (2) on S:
Uo(s) =Ea^(Pn(s). (6)
1721
Eq.(6) represents an expansion of the known function uo(s) in terms of the
functions q)„(x), non-orthogonal on S, which can be reduced in different ways
to a set of linear algebraic equations. In practice, the number of normal modes
is taken to be finite, and the number of the unknowns a„ in Eqs.(5),(6) as well
as the number of the algebraic equations is finite, too. Discretization of the
continuous operators in (l)-(3) also leads to finite systems of linear equations.
Thus, in all the above formulations the problem of the field reconstruction
reduces to the linear operator equation of the first kind
(7)
where, in theory, the functions u(s) and f(q) are elements of two functional
spaces U and F, and /4 is a linear operator. In practice, u and / are vectors
of two Euclidean spaces, and A is a rectangular matrix operator. •
General properties and formal solution
The problem of field reconstruction has the following general
properties. The problem has a bounded solution if the vector-function Uo(s) is
sufficiently smooth -see below. The solution, if it exists, is unique. It means that
the absence of information on the inaccessible part O of the boundary is
completely compensated by the overdetermined boundary conditions (2) on the
accessible part .S'. The proof of the uniqueness is based on the Almansi's
theorem[9] according to which if on a part (even very small) of the surface of a
finite elastic body displacements and stresses are simultaneously equal to zero,
then the stresses are zero everywhere and the body is at rest. Almansi proved
the theorem for statics; an extension to the dynamic theory of elasticity is given
in the paper [7]. One more property; the solution of the FR-problem does not
continuously depend on the input data: small variations A«o of the prescribed
function may cause large variations A/ of the solution. This can be concluded
from the general properties of the Fredholm integral equations of the first kind
with continuous kernels (the case of Eq.3). Hence, the problem of field
reconstruction belongs to the class of ill-posed in the sense of Hadamard
problems of mathematical physics.
Ill-posed problems are often met and solved in various scientific
disciplines. For example, the mathematical problem of differentiating functions
is ill-posed; the mechanical problem of reconstructing the external forces
through the structural response mentioned above is also ill-posed, etc.[10,ll].
All the existing methods of treating such problems are principally based on the
idea of replacing the ill-posed problem with a well-posed problem appropriately
chosen with the aid of an additional information concerning the desired
solution. For treating the field reconstruction problem, the most appropriate is
1722
the Singular Value Decomposition (SVD) technique[12]. Below follows the
solution of the FR-problem obtained by this technique.
Let aj > 02 >... be the singular values of the operator A in Equation
(7), and {fi, f2, ... } and (uj, U2, ... } be the singular pair, i.e. two sets of
. orthonormal functions (Note that of are the eigen values of the Hermitien
operators A A and AA*, while fj and Uj are their eigen functions).
Representing a given function w as a series in terms of Uj and seeking the
solution as a series in fj , one can obtain the following exact formal solution to
the problem (7):
0£3 OO
f=Z(bj /<Jj) fj , where u = Zbj Uj . (8)
It is seen from Eq.(8) that a bounded solution to the problem exists (the series
converges) if the given function w is sufficiently smooth or, more exactly, if
the coefficients bj of its expansion (8) tend to zero more rapidly than the
singular values q;- . For real structures, the singular values tend to zero very fast
(exponentially), therefore the exact functions of distribution of the vibration
amplitudes at the accessible part S of the surface should be and actually are
very smooth and do not contain components rapidly oscillating along the
surface, i.e., addenda in (8) with large numbers j. However, small but finite
random errors in u, experimental or due to rounding in a computer, have wide
spatial spectra and result in expansions (8) with non-zero coefficients bj of large
numbers which, after enhancing by the small singular values, may cause large
errors in the solution f, making it unstable.
The simplest way to overcome the difficulty is to seek an approximate
solution which involves a finite number N of terms in the singular value
decomposition, i.e. to truncate the series’ in (8). With a judicious choice of N
this solution does not differ too much from the exact solution (8), because the
exact (not contaminated by noise) vibration function u does not contain
components with high numbers. On the other hand, this solution cuts off the
rapidly oscillating components j >N, thereby reducing the inaccuracy due to
experimental or computer errors in the input data. Truncating the series (8), we
restrict the solution by smooth functions and this is the additional a priori
information which makes the problem well-posed. However, the choice of the
best truncation number is a difficult and unstudied question, and answering
it was one of the objectives of the computer simulation and laboratory
experiment.
Results of computer simulation
In computer simulation, two simple structures were studied - a finite
longitudinally vibrating rod and a thin elastic strip executing vibrations in its
1723
plane(Fig.2).The rod(Fig.2a)
is excited by a harmonic
force at the left end. The
accessible for measurement
part is supposed to be the
region S at the right end
(data region), while the rest
part 0 of the rod is
considered as the
reconstruction region. The
complex displacement
amplitude u(x) satisfies the
Bernoulli’s equation of
longitudinal vibrations [13]
with a complex Young
modules. This boundary
value problem has an
analytical solution which has
been used for computing the
input data “measured” in the region S and for estimating the accuracy of
reconstruction in the region O. In calculations, a modal model was used with
the modes corresponding to the free boundary conditions at both ends:
{ u„(x) = (s„/ if ^ cos (7t(n -i)x/l); Sj =1, Sn=2 for n = 2,3,...}.
The solution (5) has here the form of a finite sum
A/
u(x) ^ (9)
with unknown amplitudes a„ . Equating the representation (9) to M measured
values of u a.t M points of the region S gives a set of M linear algebraic
equations with N unknowns that were solved by the SVD-technique.
The elastic strip (Fig. 2b) of height 2H comparable with the elastic
wavelengths vibrates in its plane. It is supposed that only upper surface y =H
of the strip is accessible for measurement. It is required to reconstruct the
vibration field in the region V with the following dimensions: / =3H.
Vibrations of the strip satisfy the dynamic equations of the plane stress state and
the boundary conditions of absence of stresses at y =-H and y H. As it is
known, they are analogous to vibrations of an elastic layer. Therefore, the field
in the strip is represented as a sum of N Lamb’s normal modes [13]:
A//2
{U:,(x,y): Uy(x,y)} = Z[a„ {u^'"(y): Uyn(y)} + (10)
b„ {u,cn (y): Uy„-(y)} tx^(ik„(l-x))],
where the K are the propagation constants, i.e., the roots of the Rayleigh-
Lamb dispersion equation, and the expressions in brackets describe the mode
/ -
0
y
H
- ^ (a)
/ X
/ '
i 1
\l X \
-H
Fig.2. Two simp
simulation: a loi
and an elastic si
0
le structures used in computer
ngitudinally vibrating rod (a)
tip vibrating in its surface(b).
1724
forms. On the accessible surface jl/ =H of the strip, M/2 equidistant points are
chosen at which the displacement components, Ux and are computed
using the exact solution. (The exact solution is taken as in an infinite strip
excited by a unit y-force applied to the point (-4H, H), i.e., at the distance 4H
from the region of interest V). Equating the “measured” amplitudes to (10)
gives a set of M linear algebraic equations with N unknown mode
amplitudes, and h„ , which can be found by the SVD-technique.
The results of computer simulation obtained for the two different
structures on Fig. 2 (the first structure is one-dimensional and the second is two-
dimensional) are similar. They are also similar to those obtained for other
structures (the author verified them for an acoustic waveguide and for a
circular cylindrical shell). So, they are rather general.
The first and practically most important result is the existence of an
optimal model: there is a number No of normal modes (model -parameters)
which renders minimum to the reconstruction error. It means that too
complicated and exact models containing an excessive number of model
parameters N > No as well as rough models with a small number of the
parameters N < No, give larger errors of field reconstruction than the optimal
model with No parameters. The best number No depends on the structure
type, geometry, frequency, etc., but most of all on the accuracy and amount of
Fig. 3 presents the
field reconstruction
error A versus the
number of the model
parameters
computed for the
strip on Fig.2b. The
error A is defined
as relative square
module deflection of
the reconstructed
displacement from
the exact one
averaged over the
reconstruction
region V. In
computing these
plots, a random
error of prescribed
standard value S is
added to the
“measured” data. Four curves in Fig,3 correspond to various values of the
1725
standard error 5. It is seen, that each curve of the field reconstruction error has
a pronounced minimum beyond which the error sharply increases reaching
arbitrary large values. This result seeming paradoxical has a clear physical
explanation. The reason lies in random errors of the input data.
If there were no input errors, i.e., if the input data were known mathematically
exactly, the reconstruction error would monotonically decrease with the
number of modal parameters N: the more exact is the model the better is the
approximation. When input errors are introduced, the decreasing of A(N)
holds only to a certain limit since for large N the errors in the input data,
enhanced by small singular values (see equation 8), may give an arbitrarily
large error in the result of reconstructing the field.
. 15
Fig. 4.Amplitude(a) and phase(b) ofthe bardisplacement
exact -solid lines, reconstructed - dotted lines: kl=4. 7;
loss factor 0. 05; number of modes used 2. _
Thus, for each
value of the input
error, there exists
an optimal number
of the model
parameters which
corresponds to the
minimal error in the
reconstructed field.
Curves, similar to
the curves in Fig.3,
are obtained also
for other structures.
For example, when
30% of the bar in
Fig.2a are
accessible for
measurement and
the input error is
equal to 10*^, the
best number of
modes is No ""12.
Fig.4-6 present the
displacement
amplitude and
phase distributions
along the rod for a
rough model {N
=2), the optimal
model (N=J2) and
an excessively
complicated model
1726
(N= 30), It is seen
that in all cases the
reconstruction
error, being small in
the measurement
region S, increases
with distance from
Another finding
in the computer
simulation concerns
the amount of input
data necessary for
obtaining the best solution. Fig.7 presents a typical dependence -of the field
reconstruction error on the number M of measurement points in the region S
for alongitudinally vibrating bar (Fig.2a).
It is seen, that the reconstruction error is
unstable when the number M is small.
With increasing M the error A
becomes stable and decreases tending to
a certain constant value. Similar
dependencies take place for the strip and
other structures. The optimal amount of
the input data correspond to the number
of measurements M two or three times
the number N of the model parameters
involved. A further increase of the
amount of data does not improve the
results and is, therefore, unjustified.
It should be noted that all
components of the prescribed displacement vector are needed in order to obtain
the best reconstruction error. This follows from the uniqueness theorem
mentioned above: the problem of field reconstruction is uniquely solvable only
if the full displacement vector on the accessible surface is available.
Some experimental results
To verify the proposed method, a laboratory experiment has been
carried out on a circular cylindrical shell (Fig. 8). A finite open shell with
dimensions in mm 900x300x3.5 is excited by a shaker with harmonic signals.
The amplitudes and phases of vibration are measured at the surface 5' by a
small (2g) 3-component accelerometer, the data are fed into a PC. The
1727
amplitudes and phases of vibration in the region O are reconstructed (by the
algorithm described above) and compared with the actual ones. The
reconstruction error is computed with respect to the actual field of the region
0. In modelling the vibration field in the shell, the displacement vector is
represented by the series of the normal modes which satisfy the simplest shell
equations (the Donnel-Mushtari theory [14]). The maximal number of the
normal modes used is 52. These modes correspond to the circumferential
numbers up to m =10, The evanescent modes decaying more than 40 dB at the
distance from the region S+Q to the end of the shell or to the shaker, are
excluded from the model. The total error of measuring complex vibration
amplitudes is estimated as 0.07 < S< 0.08.
Fig. 9 and 10
show the dependencies
of the reconstruction
error A (solid lines)
on the complexity of
models used for one of
the frequencies (f =
700 Hz) and for two
different amount of
input data. The curves
are obtained as
follows.
First, the simplest
models containing the
normal modes of one
single circumferential
1728
number m are tried.
Among 1 1 models
with m = 0,1,..., 10,
the best model is
chosen which gives
the minimal error in
describing the
measured data (in
practice, only these
data are actually
available). In both
cases of Fig. 9, 10 it is
the model with the
circumferential
number m ^ 6. It
corresponds to the
model complexity MC = 1 in Fig. 9, 10. Then, all models with two
circumferential numbers (MC = 2 in Fig.9,10j are considered: one is mj = 6
and another is chosen among 10 models with m2 ^6 which gives the minimal
error in describing the input data (It is found that m2 equals 5 in the case of
Fig.9 and m2 ^4 for Fig. 10). After that, all models with three circumferential
numbers (MC=3) are investigated, etc., until all the models are exhausted.
Thus, the dashed curves in Fig.9, 10 correspond to the best models of various
complexity for the measurement region S of the shell. Therefore, these curves
decrease monotonically, at least, do not increase, with the number of the model
parameters. At the same time, the error of the expansion to the reconstruction
region 0 (solid curves in Fig.9, 10) has tendency to increase with the number
of the model parameters. More exactly, they have minima beyond which they
increase monotonically. For very complicated models, the reconstruction error
may reach hundred of thousand. With these curves, correlate the curves of the
condition number (dotted lines).
Now, the question how to choose the optimal model which minimises
the reconstruction error in the inaccessible region (i.e., corresponds to the
minimum of the solid line curves in Fig.9, 10) using only the information about
vibration of the measurement (accessible) region (i.e., using the dashed and
dotted line curves in Fig.9, 10) may be answered as follows. As one can observe
from Fig.9 and 10, the error of describing the measured data (dashed lines)
first rather rapidly decreases with the model complexity and then becomes
almost constant. The best model just corresponds to the transition from the
interval of rapid decrease to the interval of stable values of the dashed line
curve. Conversely, the curves of the condition number (dotted lines in Fig. 9, 10)
first grow slowly with MC but, after the best model is reached, their growth
becomes rather fast.
1729
Thus, it can be concluded that the optimal model can be approximately
identified from the behaviour of the curves of the error in the input data and of
the condition number. From the physical point of view, this “rule of thumb” is
obvious - see explanation of Fig.3. But quantitatively, it is rather uncertain: one
can only find an interval of models within which the best model lies. E.g., for
the cases of Fig.9,10 the intervals correspond to MC ~ {4,5,6} and {2,3,4}
where the reconstruction error equals {0.35, 0.25, 1.2} and {0.64, 0.75, 0.85}
while the minimal values in these cases are 0.25 and 0.51. Much more certain
identification of the best model can be made if at least one measurement point
is taken in the reconstruction region; in other words, if it is possible to place at
least one sensor into the needed part of the structure. In this case, however,
another procedure of the optimal model identification is required.
Summary
The problem of expanding the vibration field from the measurement
surface to the volume of an elastic structure is rigorously posed and studied
theoretically and experimentally. When formalised, this problem, also called as
the field reconstruction problem, is a non-traditional boundary value problem
for differential equations of elliptic type. The conditions for existence of a
solution are found and the uniqueness theorem is proven. A general solution
based on the generalised singular value decomposition is obtained.
Results of computer simulation with simple structures are presented.
Most attention is paid to studying the accuracy of the expansion to unmeasured
parts of the structure. Relations between the expansion accuracy, the
measurement errors and the complexity of the vibrational model of the
structure (number of model parameters) are established. A salient feature of
these relations is that, for a given input data accuracy, there is an optimal model
which minimises the expansion error.
Results of laboratory experiments with a steel cylindrical shell executing
forced harmonic vibration are also presented aiming to verify the obtained
relations. Procedures of choosing the optimal models are discussed.
REFERENCES
1. Ramakumar, R., Reliability engineering. Fundamentals and applications.
Prentice Hall, Allyn & Bacon and Ellis Horwood,1993,
2. Structural intensity and vibrational energy flow (Proc.4th Int. Congress on
intensity technique). Senlis, France, 1993.
3. Tyrer, J.R., Determination of surface stresses and velocities by optical
measurement. In [2], 35-45.
1730
4. Maynard, J.D., Williams, E.G. and Lee,Y., Nearfield acoustic holography
(NAH). JAcoust Soc. Am., 1985, 78, 1395-1413.
5. To,W.M. and Ewins, D.L., The role of the generalised inverse in structural
dyna.mics. J. Sound and Vibr., 1995, 186, 185-195.
6. Yen, C.S. and Wu, E., On the inverse problem of rectangular plates
subjected to elastic impulse. J. Appl Meek, 1995, 62, 692-705.
7. Bobrovnitskii, Yu.L, The problem of field reconstruction in structural
intensimetry; statement, properties, and numerical aspects. Acoust. Phys.,
1994, 40,331-9.
8. Preiss, A.K., Evaluation of stress fields by a finite body of experimental
'\nformd.i\on. Mashinovedenie, 1984, N2, 77-83.
9. Almansi, E., Un teorema sulle deformazioni elastiche dei solid! isotropi, Atti
R.Accad.Lincei, 1907, Ser.5, 16, 865-8.
10. Lavrentiev, M.M., Some improperly posed problems in mathematical
physics. Springer, Berlin, 1967.
11. Tikhonov, A.N. and Arsenin, V.Ya., Solution of ill-posed problems.
Simon&Schuster, Washington DC, 1977.
12. Golub, G.H. and van Loan, C.F., Matrix computations. North Oxford
Academic Publishing, Oxford, England, 1983.
13. Graff, K.F., Wave motion in elastic solids. Clarendon Press, Oxford, 1975.
H.Leissa, A.W., Vibration of shells. US Government Printing Office,
Washington DC, 1973.
1731
EVALUATION OF THE EQUIVALENT GEAR ERROR BY
VIBRATIONS OF A SPUR GEAR PAIR
M. Amabili (*) and A. Fregolent (**)
(*) Dipartiraento di Meccanica, Universita di Ancona, Ancona, Italy
(**) Dipartimento di Meccanica ed Aeronautica, Universita di Roma
“La Sapienza”, Roma, Italy
ABSTRACT
A new approach based on the measurement of the gear torsional vibrations is
proposed to evaluate the equivalent gear error of a spur gear pair and to
identify the natural frequency and the damping of the system. The test bench is
modelled as a single degree of freedom system and must be realised by using
stiff bearings and torsionally compliant shafts. The algorithm is based on the
use of the harmonic balance method. Results can be obtained by using a quite
simple experimental apparatus. The proposed approach has some advantages
with respect to the traditional metrological methods. The effect of
measurement errors on the accuracy of the identification is also investigated.
1. INTRODUCTION
In the last decades many papers were published on the effect of gear
errors on the dynamic response of gear pairs, e.g. references [1-16]. In fact
vibrations of gear pairs are largely affected by the amplitude and phase of
deviations of the tooth profile from the true involute one. Pitch, pressure angle
and mounting (eccentricities and misalignments) errors also are of great
importance. Therefore gear errors must be checked to avoid bad working
conditions of high speed gears and silent reducers. Moreover profile
modifications are introduced to reduce gear vibrations and their accuracy must
be verified. Analytical [3, 14, 17-18], numerical [2, 9-13, 15, 19-24] and
approximate [6] methods were proposed in the past to simulate the dynamics
of a spur gear pair, and single [2-3, 5-12, 21-22], multi [4, 20, 23-24] or
infinite [25] degrees of freedom were used by different authors to model the
system’s behaviours. Multi axes reducers were investigated e.g. in references
[13-14].
In the present study an approach based on the measurement of the gear
torsional vibrations is proposed to evaluate the equivalent gear error of a spur
gear pair and to identify the natural frequency and damping of the system. The
equivalent error is a function of the gear position and is related to the errors of
the driving and the driven gears and to the non-dimensional stiffness of the
teeth; its dimension is length. The system is analytically studied by using a
single degree of freedom system capable of modelling the experimental test
1733
bench, which therefore must be realised using stiff bearings and torsionally
compliant shafts. If gear pairs have different center distances, an appropriate
housing or different housings must be built.
The algorithm is based on the use of the harmonic balance method [26]:
it is applied to spur gear pairs having low contact ratio e{le., 1 <£< 2) and is
suitable to identify pitch, profile, pressure angle and runout errors. Results can
be obtained by using a quite simple experimental apparatus requiring only the
measurement of vibration response of the driven gear during a revolution for
at least three different rotational speeds. The gear pair must be tested with a
fixed static load. When testing a single gear this must be coupled with a
reference gear.
The proposed approach presents some advantages with respect to the
metrological methods used to measure gear errors on driving and driven gears
[10]. In fact these methods, that uses control machines, provide the charts of
profile errors and cumulative pitch and runout errors for each tooth of the two
gears. On the contrary, using the proposed technique, the equivalent error is
obtained; it is directly related to the gear vibrations and therefore is
particularly appropriate to evaluate the gear accuracy from a dynamic point of
view. In fact it is well known that some modifications of the tooth involute
profile can provide a reduction of the vibration level, so that the effect of these
modifications, the accuracy of gear profiles and mounting can be checked by
using the equivalent error.
The effect of noise on the identification of the equivalent error and
modal parameters is also investigated. Some simulated tests are performed
with noise polluted vibration responses of the gear pair.
2. VIBRATION SIMULATION
A pair of spur gears is modelled with two disks coupled by nonlinear
mesh stiffness, mesh damping and excitation due to gear errors. One disk (the
driving gear) has radius and mass moment of inertia /i, while the other (the
driven gear) has radius i?2 and mass moment of inertia h', the radii R\ and R2
correspond to the radii of the base circles of the two gears, respectively.
The transmission error, defined as the difference between the actual and
ideal positions of the driven gear, is expressed as a linear displacement along
the line of action. The sign convention used for the transmission error is
positive behind the ideal position of the driven gear. Analysing gears with low
contact ratio e (i.e., 1 < £ < 2), the nonlinear equation of motion for the
dynamic transmission error jc can be written as;
mx + cjc + Z, (j:,r) + /2(A:,r) = Wq, (1)
where
J!C= R, 0, ^2’ (^)
being B\ and 62 the angular displacements of the two gears; the equivalent
inertia mass m of the system is:
1734
(3)
Wo is the static load given by
W,^TJR,=T,IR^. (4)
being T\ and T2 the driving and driven torques, respectively; fj (;c, f) are the
elastic forces of the meshing tooth pair;, for; = 1,2
when x-ej(t)>0
v/hQnx-ej{t)<0.
(5)
Obviously, equation (1) can be easily extended to high contact ratio gears. In
equation (5) ki{t) and kiit) are the time-varying meshing stiffness of the two
pairs of meshing teeth. The error functions ei(f) and eiit) are the displacement
excitations representing the relative gear errors of the meshing teeth; when
two pairs of teeth come into contact there will be two separate error functions,
each acting on a different spring. It is assumed that positive error functions
give a positive transmission error. Error functions represent the sum of pitch,
profile, pressure angle and runout errors. Moreover, when separation of tooth
pairs occurs, because of the relative vibrations and backlash between the gear
teeth, the dynamic forces;^- (x, t) are zero, according to equation (5); these are
the nonlinear terms in the equation of motion. In equation (1) a constant
viscous damping is assumed.
The total stiffness of the gear pair is given by k(t) = ki{t)+k2{t). Let us
introduce the meshing circular frequency co = z^2 , where Q is the angular
velocity of the driven gear [rad s‘‘] and z its number of teeth. The stiffness
kj{t), which is a periodic function, has a principal period T-27t/co. The
behaviour of kj{t) and k{t) are discussed, e.g., in references [3, 6, 10] and in
section 4.
Now the case when x-ej{t) > 0 is considered, Le., when there is contact
between the two gears. Therefore the following study is correct when there is
no tooth separation between driving and driven gears. The phenomenon of
tooth separation is described, e.g., in [13]. The equation of motion (1) can then
be written as a second order linear ordinary differential equation;
x + 2^C0qX + C0q K{t) x=Fq+ cOq K, (0 e, (0 + col {t) (r) , (6)
where: K{t) = k{t)lk„^, K^{t)-k^{t)lk^, K^{t) = k^{t)lk„^, Fo=Wo/m and f
is the damping ratio; K, K\ and K2 are non-dimensional functions. Moreover
the average value of the mesh stiffness is = (i/t) j/:(r)df and
0
is the natural circular frequency of the undamped system with stiffness equal
to its integral average value.
It is useful to introduce the following Fourier expansion of the
equivalent error [m]
1735
(7)
K,{t)e,(t) + K^(f)e^(.t)= ,
n®-®®
where i is the imaginary unit. The expression on the left side of equation (7)
represents the equivalent error of the gear pair; in fact, it is the excitation due
to gear errors on the right side of equation (6). It is assumed that this function
has a principal period zT, Le. the time revolution of the driven gear, and
therefore has principal circular frequency Q. Strictly the same tooth of the
driving and driven gears mesh together only after a period s\zT, where and
$2 are the integers that express the gear ratio ras the rational number
(usually T < 1). In this study the error components having circular frequency
lower than Q are neglected. It is important to observe that in many cases the
profile errors can be considered the same for all the teeth of the gear, i.e. they
all have principal period T\ therefore the coefficients dn for n = z, 2z, 3z, ... are
due to periodic profile errors, whereas the others are due to pitch, pressure
angle and runout errors.
The expansion of the non-dimensional total meshing stiffness is
a:(i) . (8)
The solution of the equation of motion (6) is obtained by using the harmonic
balance method. Therefore the dynamic transmission error x is expanded into
a complex Fourier series
(9)
n--t»
Substituting equations (7-9) into equation (6) the following equation is found
= fo+£Bo
Equation (10) gives the following algebraic linear system
AC = F, (11)
where the elements of the matrix A are given by
, (12)
fl if {n-j)/z isintegerl
On / is the Kronecker delta and W„ ^ , . r • Moreover it
[O otherwise J
is:
1736
(13-14)
0)1
cold,
F = ^
Fo+<old„
^-1
cold.,
. (old.^ .
3. IDENTIFICATION OF MODAL PARAMETERS AND
GEAR ERRORS
The aim of this work is to identify the equivalent gear error; therefore
the vector F in equation (11) is now unknown. On the contrary the contact
ratio £, the shape of the stiffness function Kit) and all the constants of the
expansion are known. The transmission error x(r) is then experimentally
measured for different rotational speeds Q of the driven gear; in particular,
only rotational speeds where no tooth separation occurs must be chosen. It is
obvious that the dynamic transmission error varies according to the speed Q,
so that the constants c„ of the expansion and the vector C are functions of O.
Equation (12) shows that also the matrix A is a function of JQ; whereas the
vector F is independent of it. Therefore it can be written
F = A(I2)C(I2). (15)
In equation (15) the vector C(I2) is obtained experimentally and the matrix
A(I2) theoretically by using equation (12). However in order to compute the
matrix A the modal parameters coq and f of the system must be identified
because they appear in equation (12). These parameters can be determined by
using the following equation:
A (12, ) C(^2i ) = A{£2^) C(jQ2 ) = A (12. ) C(I2,. ) = constant , (16)
where A are fixed rotational speeds. Then
,r
X, I ' c,. + 2 in f a)„ c„ + aj
= constant .
(17)
Computing the left hand side of equation (17) for different rotational speeds,
e.g. I2i, ^22, 123, and subtracting first the quantity computed for Q2 to the one
computed for Q\ and then subtracting the quantity computed for to the one
computed for one obtains the following linear system
1737
(18)
£2nl[r3, c„ (i3,)-r2, c„ (A)] £ X«;[c„.^(i3,)-c„-,(A)]
^2ri[^3j C„ (^j)— (^3)] ^ £*^y [^n-;;('^l)~^ji-;/(^3)]
ml J
£«^[^2,^c,(^2,)-^23^c„(A)]
;t=-oo
%n^[Qfc„(n{)-nic^{Q,)]
The linear system (18) allows to identify the modal parameters coq and f and,
using equation (15), the vector F that gives the equivalent gear error.
Considering that usually col ^0 is negligible with respect to Fq, also the ratio
Wo/m can be identified; therefore if the static load Wo is known, the reduced
mass m of the system is obtained. Actually all the constant terms of the
identified vector F can be attributed to the static load, giving to do the zero
value. In fact the static load can be easily considered as the mean value of the
load during the gear meshing and a non zero coefficient do is equivalent to
change the static load.
In system (18) one can substitute the quantity computed for the
difference Q2-Q1 to one of the two differences previously computed. However
it is important to observe that with measurements at three different speeds
only two linearly independent equations can be obtained for the system (18).
Due to the errors introduced in the experimental measurement of the
dynamic transmission error x (errors in C) it is preferable to solve an
overdetermined system using different velocities to obtain additional
equations in system (18). Moreover the problem is ill-conditioned, so that the
errors on the known vector C are amplified in the solution. In order to
overcome this problem, it is necessary to use only the more significant
harmonics in the identification when significant measurement errors or
differences between the single degree of freedom model and the actual test
bench are observed; therefore all the sums involved in system (18) must be
stopped at an integer n not too large because higher order harmonics involved
in C only introduce noise and does not give additional information. This
process is similar to the use a low-pass filter on signals coming from sensors
used in experiments. A discussion on this phenomenon is deferred to section
5. The natural circular frequency of the system can be evaluated theoretically
or experimentally and the damping ratio can be also experimentally
determined by an impact test, using the logarithmic decrement, or can be
assumed in the range between 0.07 and 0.1, as verified by many authors [6,
10]. Therefore the results of the identification can be compared to data
obtained in a different way.
The vector F can be also determined by using equation (15) or the
following expression:
1738
F = (1//)XA(A)C(A). (19)
1=1
This procedure provides a good accuracy in the computation also in presence
of measurement errors, and the average reduces the errors that are introduced
in the experimental measurement of (and then in C).
4. APPLICATION OF THE METHOD
In the numerical simulations the stiffness function proposed by Cai and
Hayashi [6] was used; in particular the following function can be introduced
/(f) = 5;^{^^+((e-l)/2)r]V^[r+((£-l)/2)r] + 055 . (20)
The two non-dimensional meshing stiffness K\(t) and K2(t) are directly
obtained by equation (20), i.e.:
K,{t) = fit) forO<t<T (21)
if i>r-((£-i)/2)7'
’ [o if f<T-((e-l)/2)r
fit + T) if r<((£-l)/2)rl ^
if ?>((£-i)/2)r
for 0 < r < r.
The meshing stiffness is therefore a function of the contact ratio £. The two
non-dimensional meshing stiffness Ki{t) and Kzit) are shown in Figure la (€ =
1.8) along the meshing period T-27t/co, and the non-dimensional total meshing
stiffness K{t) is plotted in Figure lb. The integral average stiffness of the pair
k,n is related to the maximum stiffness of one pair of teeth /:max by the
following expression: =0.85£ ^max* ISO/DIS 6336-1.2 (1990) design
code gives a formula to evaluate kuAX-
In order to simplify the experimental measurement of the dynamic
transmission error, only a test on the driven gear can be performed. Usually
only the acceleration of the driven gear ^2 is measured; however the
acceleration x can be obtained by using the following relationship
^ ..
X, = - X, (23)
■ m, +171;,
where mi and m2 are the reduced masses of the driving and the driven gear,
respectively. The measured acceleration, can be related to the coefficients c„
by equation (9), to yield:
1739
Equation (24) shows that the coefficient cq cannot be obtained by vibration
measurement; however this coefficient can be easily determined because it is
the mean value of the dynamic transmission error x. A good estimation of this
value is Cq . In the identification process it is very important to use
only measured accelerations when no tooth separation occurs. Therefore it is
generally necessary to avoid rotational speeds larger than half the main
resonance speed coq.
The benchmark for gears can be obtained by a variable speed motor and
a brake. Torsional flexible shafts, with a natural frequency lower than 1/10 of
the natural frequency Q)o of the gear pair, and stiff bearings must be used in
order to well approximate a single degree of freedom system. The rotational
speed can be measured by a proximity sensor that counts the number of
passing teeth; the load is measured by a dynamometer and the acceleration of
the driven gear by accelerometers. In particular two identical accelerometers
which are mounted on the gear through two small aluminium blocks are used;
they can be placed on opposite sides of a diameter and must have tangential
orientation. A summing amplifier sums up the signals from the accelerometers
to eliminate the effect of eccentricity. A slip ring is introduced to bring the
signals to the amplifier. In alternative a laser rotational vibrometer can be
employed to measure vibrations of the driven gear; in this case accelerometers,
slip ring and summing amplifier can be eliminated. Lubrication of the meshing
teeth must be provided.
5. NUMERICAL RESULTS
To verify the proposed method the gear pair tested by Umezawa et al.
[10] is studied. The two gears are finished by a MAAG grinding; the
characteristics of both the gears are: module = 4, number of teeth = 48, face
width = 10 mm, pressure angle = 14.5®, diameter of standard pitch circle =
192 mm; the gear ratio is 1. The contact ratio e is 1.8, the rotational speed
range Q, - 400 ^ 3000 rpm (41.89 -4- 314.16 rad s’^), the torque 196 Nm and
the teeth have involute profile. The profile errors are about bjim at the root of
the driving gear and at the tip of driven gear; these profile errors are reported
in reference [10]. The natural circular frequency is coo = 48x3062 rpm (2450
Hz) and the value of the damping ratio f is 0.07.
The response of the gear pair is polluted with noise on the gear data and
modal parameters. It is assumed that only profile errors are significant in this
case, so that all the other errors are neglected. As a consequence of this
hypothesis, the dynamic transmission error has a principal period equal to the
meshing period T. The response x is discretized with 201 points in the period.
Then the noise is added to the response before computing the vector C. In
particular the noise is generated using random numbers added to the time
response. These random numbers are obtained by a normal distribution having
zero mean value and variance G~ lev* max, where max is the maximum value
of the response x in the period and lev is the error level. As a consequence of
1740
the assumed distribution, the 68 % of the points have noise within ic7, the 95
% within ±2(7 and the 99.7 % within ±3(7.
First, equation (18) is used to identify the modal parameters of the
system from noise polluted responses. In particular eight responses at
rotational speeds Q. = 60, 65, 68, 70, 80, 90, 100, 120 rad s*' with an error
level /ev = 0.02 are employed. This error level gives responses having a
difference within ±6% with respect to the true value. In Figure 2a the percent
difference between the identified natural frequency GOo and the actual value is
plotted versus the number of harmonics used in the identification. The range
of harmonics of the meshing frequency co=z^ that gives correct results is
4^<8. In Figure 2b the data relative to the damping ratio f are reported.
Figure 3 is similar to Figure 2 but it is obtained for an error level lev = 0.005
(difference within ±2.5% with respect to the true value). In this case the range
of useful harmonics increases. Figure 2 and 3 show that it is not convenient to
exceed in the number of harmonics. In fact, for the considered problem, the
amplitude of coefficients c,j decreases with n so that higher order harmonics
are largely affected by the noise. Being the identification process an ill
conditioned problem, it is necessary to employ only harmonics having a good
signal to noise ratio.
In Figure 4 the equivalent error obtained by using the profile errors
given in reference [10] and equations (7), (20-22) is shown; this one can be
called the “actual” equivalent error. In Figure 5 the equivalent error is
identified by using a response at = 60 rad s'^ without noise. In this case 15
harmonics are used to describe the function. The difference between Figures 4
and 5 can be surely attributed to the truncation error.
Then eight responses at rotational speeds speeds ^2 = 60, 65, 68, 70, 80,
90, 100, 120 rad s'‘ with an error level /ev = 0.005 are used to identify the
equivalent error; the result is given in Figure 6 where the actual modal
parameters (cut) and Q are used. Figures 5 and 6 are very similar and they well
describe the “actual” error reported in Figure 4. Also in the case of an error
level of ^ev = 0.02 and using the actual modal parameters a quite good
evaluation of the equivalent error is reached, as shown in Figure 7.
The effect of an incorrect identification of the modal parameters coo and
^ on the evaluation of the equivalent gear error is then investigated in Figure
8; in this case an error of +10% on the frequency and +40% on the damping
ratio is used to evaluate the equivalent error by responses having a noise level
lev ~ 0.02 . Also in this case a quite good result is obtained.
6. CONCLUSIONS
The reconstruction of the equivalent gear error by acceleration
measurement of the driven gear of a spur gear pair is quite good also in the
case of measurements affected by noise. In particular the identification of the
natural frequency and damping of the system is obtained by an overdetermined
1741
linear system to minimise the error; to obtain correct results only lower order
harmonics must be considered. Then these data can be used to evaluate the
equivalent gear error that is not largely affected by the inaccuracy in the
identification of the natural frequency and damping.
The proposed method seems to have advantages in the quality control of
a large production of gears having the same center distance; in this case the
same housing can be used. Moreover the gear can be tested together with a
reference gear or with the companion gear to check the actual gear pair. As a
limitation, the method can be only applied to systems having stiff bearings and
torsionally compliant axes, requiring an appropriately designed experimental
apparatus.
REFERENCES
1. H. N. OZGUVEN and D. R. HOUSER 1988 Journal of Sound and Vibration 121,
383-41 1 . Mathematical models used in gear dynamics.
2. H. N. OZGUVEN and D. R. HOUSER 1988 Journal of Sound and Vibration 125,
71-83. Dynamic analysis of high speed gears by using loaded static transmission
error.
3. M. AMABILI and A. RiVOLA to appear in Mechanical Systems and Signal
Processing. Dynamic analysis of spur gear pairs: steady state response and
stability of the SDOF model with time-varying meshing stiffness.
4. P. VELEX and M. MaaTAR 1996 Journal of Sound and Vibration 191, 629-660.
A mathematical model for analyzing the influence of shape deviations and
mounting errors on gear dynamic behaviour.
5. Y. Cai and T. Hayashi 1992 Proceedings of the ASMS International Power
Transmission and Gearing Conference, Scottsdale, Arizona 2, 453-460 (DE -
Vol. 43-2). The optimum modification of tooth profile for a pair of spur gears to
make its rotational vibration equal zero.
6. Y. Cai and T. HayaSHI 1994 Transactions of the American Society of
Mechanical Engineers, Journal of Mechanical Design 116, 558-564. The linear
approximated equation of vibration of a pair of spur gears (theory and
experiment).
7. Y. Cai 1995 Transactions of the American Society of Mechanical Engineers,
Journal of Mechanical Design 117, 460-469. Simulation on the rotational
vibration of helical gears in consideration of the tooth separation phenomenon
(a new stiffness function of helical involute tooth pair).
8. Y. Cai 1996 Proceedings of the ASMS International Power Transmission and
Gearing Conference, San Diego, CA, 177-184 (DE - Vol. 88). Development of a
silent helical gear reducer (vibration simulation and noise measurement).
9. T. Sato, K. UMEZAWA and J. ISHIKAWA 1983 Bullettin of the Japan Society of
Mechanical Engineers 26(221), 2010-2016. Effects of contact ratio and profile
correction on gear rotational vibration.
10. K. UMEZAWA, T. Sato and J. ISHIKAWA 1984 Bullettin of the Japan Society of
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spur gears.
11. K. UMEZAWA, T. Sato and K. KOHNO 1984 Bullettin of the Japan Society of
Mechanical Engineers 21{225), 569-575. Influence of gear error on rotational
1742
vibration of power transmission spur gears (1st report, pressure angle error and
normal pitch error).
12. K. UMEZAWA and T. SATO 1985 Bullettin of the Japan Society of Mechanical
Engineers 28(243), 2143-2148. Influence of gear error on rotational vibration of
power transmission spur gears (2nd report, waved form error).
13. K. UMEZAWA, T. AJIMA and H. HOUJOH 1986 Bullettin of the Japan Society of
Mechanical Engineers 29(249), 950-957. Vibration of three axes gear system.
14. A. KUBO, S. KIYONO and M. FUJINO 1986 Bullettin of the Japan Society of
Mechanical Engineers 29(258), 4424-4429. On analysis and prediction of
machine vibration caused by gear meshing (1st report, nature of gear vibration
and the total vibrational excitation).
15. R. KasuBA and J. W. Evans 1981 Transactions of the American Society of
Mechanical Engineers, Journal of Mechanical Design 103, 398-409. An
extended model for determining dynamic loads in spur gearing.
16. W. D. Mark 1987 Transactions of the American Society of Mechanical
Engineers, Journal of Mechanisms, Transmissions, and Automation' in Design
109, 268-274. Analytical reconstruction of the running surfaces of gear teeth.
Part 2: combining tooth spacing measurements with profile and lead
measurements.
17. M. Benton and A. Seireg 1978 Transactions of the American Society of
Mechanical Engineers, Journal of Mechanical Design 100, 26-32. Simulation
of resonances and instability conditions in pinion-gear systems.
18. R. W. Cornell and W. W. WESTERVELT 1978 Transactions of the American
Society of Mechanical Engineers, Journal of Mechanical Design 100, 69-76.
Dynamic tooth loads and stressing for high contact ratio spur gears.
19. B. M. BahgaT, M. O. M. Osman and T. S. SankaR 1983 Transactions of the
American Society of Mechanical Engineers, Journal of Mechanisms,
Transmissions, and Automation in Design 105, 302-309. On the spur-gear
dynamic tooth-load under consideration of system elasticity and tooth involute
profile.
20. A. S. Kumar, T. S. SaNKAR and M. 0. M. Osman 1985 Transactions of the
American Society of Mechanical Engineers, Journal of Mechanisms,
Transmissions, and Automation in Design 107, 54-60. On dynamic tooth load
and stability of a spur-gear system using the state-space approach.
21. D. C. H. Yang and Z. S. SUN 1985 Transactions of the American Society of
Mechanical Engineers, Journal of Mechanisms, Transmissions, and Automation
in Design 107, 529-535. A rotary model for spur gears dynamics.
22. D. C. H. Yang and J. Y. LiN 1987 Transactions of the American Society of
Mechanical Engineers, Journal of Mechanisms, Transmissions, and Automation
in Design 109, 189-196. Hertzian damping, tooth friction and bending elasticity
in gear impact dynamics.
23. H-H. LiN, R. L. HUSTON and J. J. COY 1988 Transactions of the American
Society of Mechanical Engineers, Journal of Mechanisms, Transmissions, and
Automation in Design 110, 221-225. On the dynamic loads in parallel shaft
transmissions: Part I - modelling and analysis.
24. H-H. Lin, R. L. Huston and J. J. Coy 1988 Transactions of the American
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Automation in Design 110, 226-229. On the dynamic loads in parallel shaft
transmissions: Part II - parameter study.
1743
25. O. S. §ENER and H. N. OZGUVEN 1993 Journal of Sound and Vibration 166,
539-556. Dynamic analysis of geared shaft systems by using a continuous
system model.
26. A. H. Nayfeh and D. T. MOOK 1979 Nonlinear Oscillations. New York: John
Wiley and Sons.
1.3
1.2
1.1
1
0.9
0.8
0.7
0,6
0.5
0 T
t
Figure 1. (a) The non-dimensional meshing stiffness Ki(t) and Kzit). (b) The
total meshing stiffness K{t).
t
number of harmonics
number of harmonics
Figure 2. Results of the identification with an error level /ev=0.02 (a) Percent
difference between the identified coq and the actual value V5. the number of
harmonics used in the identification, (b) Percent difference between the
identified f and the actual value v^. the number of harmonics.
1744
difference [%]
number of harmonics
number of harmonics
Figure 3. Results of the identification with an error level /ev=0.005 (a) Percent
difference between the identified and the actual value, (b) Percent
difference between the identified f and the actual value.
0
Figure 4. Actual equivalent gear error.
t
Figure 5. Identified equivalent gear error by response without noise.
1745
0 T
t
Figure 6. Identified equivalent gear error by responses with
/5v=:0.005; actual modal parameters.
0 T
t
Figure 7. Identified equivalent gear error by responses with
/ev=0.02; actual modal parameters.
0 T
Figure 8. Identified equivalent gear error by responses with
;ev=0.02; +10% of the actual value of coo and +40% of the actual
a noise level
a noise level
a noise level
value of
1746
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